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155-398: A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry . The corners, also called vertices , are zero- dimensional points while the sides connecting them, also called edges , are one-dimensional line segments . A triangle has three internal angles , each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals

310-415: A . {\displaystyle q_{a}={\frac {2Ta}{a^{2}+2T}}={\frac {ah_{a}}{a+h_{a}}}.} The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when a 2 = 2 T {\displaystyle a^{2}=2T} , q = a / 2 {\displaystyle q=a/2} , and the altitude of the triangle from

465-403: A {\displaystyle a} , h a {\displaystyle h_{a}} from the side a {\displaystyle a} , and the triangle's area T {\displaystyle T} are related according to q a = 2 T a a 2 + 2 T = a h a a + h

620-441: A ) ( s − b ) ( s − c ) . {\displaystyle T={\sqrt {s(s-a)(s-b)(s-c)}}.} Because the ratios between areas of shapes in the same plane are preserved by affine transformations , the relative areas of triangles in any affine plane can be defined without reference to a notion of distance or squares. In any affine space (including Euclidean planes), every triangle with

775-495: A crest , supporters , and other heraldic embellishments. The term " coat of arms " technically refers to the shield of arms itself, but the phrase is commonly used to refer to the entire achievement. The one indispensable element of a coat of arms is the shield; many ancient coats of arms consist of nothing else, but no achievement or armorial bearings exists without a coat of arms. From a very early date, illustrations of arms were frequently embellished with helmets placed above

930-453: A pseudotriangle . A pseudotriangle is a simply-connected subset of the plane lying between three mutually tangent convex regions. These sides are three smoothed curved lines connecting their endpoints called the cusp points . Any pseudotriangle can be partitioned into many pseudotriangles with the boundaries of convex disks and bitangent lines , a process known as pseudo-triangulation. For n {\displaystyle n} disks in

1085-561: A spherical triangle or hyperbolic triangle . A geodesic triangle is a region of a general two-dimensional surface enclosed by three sides that are straight relative to the surface ( geodesics ). A curvilinear triangle is a shape with three curved sides, for instance, a circular triangle with circular-arc sides. This article is about straight-sided triangles in Euclidean geometry, except where otherwise noted. Triangles are classified into different types based on their angles and

1240-426: A straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region . Sometimes an arbitrary edge is chosen to be the base , in which case the opposite vertex is called the apex ; the shortest segment between the base and apex is the height . The area of a triangle equals one-half the product of height and base length. In Euclidean geometry , any two points determine

1395-501: A bright violet-red or pink colour; and carnation , commonly used to represent flesh in French heraldry. A more recent addition is the use of copper as a metal in one or two Canadian coats of arms. There are two basic types of heraldic fur, known as ermine and vair , but over the course of centuries each has developed a number of variations. Ermine represents the fur of the stoat , a type of weasel, in its white winter coat, when it

1550-436: A circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides. Furthermore, every triangle has a unique Steiner circumellipse , which passes through the triangle's vertices and has its center at the triangle's centroid. Of all ellipses going through the triangle's vertices, it has the smallest area. The Kiepert hyperbola is unique conic that passes through

1705-520: A circular triangle whose sides are all convex. An example of a circular triangle with three convex edges is a Reuleaux triangle , which can be made by intersecting three circles of equal size. The construction may be performed with a compass alone without needing a straightedge, by the Mohr–Mascheroni theorem . Alternatively, it can be constructed by rounding the sides of an equilateral triangle. A special case of concave circular triangle can be seen in

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1860-437: A corresponding triangle in a model space like hyperbolic or elliptic space. For example, a CAT(k) space is characterized by such comparisons. Polygon In geometry , a polygon ( / ˈ p ɒ l ɪ ɡ ɒ n / ) is a plane figure made up of line segments connected to form a closed polygonal chain . The segments of a closed polygonal chain are called its edges or sides . The points where two edges meet are

2015-399: A dark red or mulberry colour between gules and purpure, and tenné , an orange or dark yellow to brown colour. These last two are quite rare, and are often referred to as stains , from the belief that they were used to represent some dishonourable act, although in fact there is no evidence that this use existed outside of fanciful heraldic writers. Perhaps owing to the realization that there

2170-553: A division of the field, which is partly metal and partly colour; nor, strictly speaking, does it prevent a field from consisting of two metals or two colours, although this is unusual. Furs are considered amphibious, and neither metal nor colour; but in practice ermine and erminois are usually treated as metals, while ermines and pean are treated as colours. This rule is strictly adhered to in British armory, with only rare exceptions; although generally observed in continental heraldry, it

2325-494: A given perimeter, the one with the largest area is regular (and therefore cyclic). Many specialized formulas apply to the areas of regular polygons . The area of a regular polygon is given in terms of the radius r of its inscribed circle and its perimeter p by This radius is also termed its apothem and is often represented as a . The area of a regular n -gon in terms of the radius R of its circumscribed circle can be expressed trigonometrically as: The area of

2480-407: A grant of arms; it may be assumed without authority by anyone entitled to bear arms, together with mantling and whatever motto the armiger may desire. The crest, however, together with the torse or coronet from which it arises, must be granted or confirmed by the relevant heraldic authority. If the bearer is entitled to the ribbon, collar, or badge of a knightly order, it may encircle or depend from

2635-464: A late use of heraldic imagery has been in patriotic commemorations and nationalistic propaganda during the nineteenth and early twentieth centuries. Since the late nineteenth century, heraldry has focused on the use of varied lines of partition and little-used ordinaries to produce new and unique designs. A heraldic achievement consists of a shield of arms , the coat of arms, or simply coat, together with all of its accompanying elements, such as

2790-409: A number is usually displayed only in documentary contexts. The Scottish and Spanish traditions resist allowing more than four quarters, preferring to subdivide one or more "grand quarters" into sub-quarters as needed. The third common mode of marshalling is with an inescutcheon , a small shield placed in front of the main shield. In Britain this is most often an "escutcheon of pretence" indicating, in

2945-525: A number of seals dating from between 1135 and 1155 appear to show the adoption of heraldic devices in England, France, Germany, Spain, and Italy. A notable example of an early armorial seal is attached to a charter granted by Philip I, Count of Flanders , in 1164. Seals from the latter part of the eleventh and early twelfth centuries show no evidence of heraldic symbolism, but by the end of the twelfth century, seals are uniformly heraldic in nature. One of

3100-401: A number of ways, of which the simplest is impalement : dividing the field per pale and putting one whole coat in each half. Impalement replaced the earlier dimidiation – combining the dexter half of one coat with the sinister half of another – because dimidiation can create ambiguity between, for example, a bend and a chevron . "Dexter" (from Latin dextra , "right") means to

3255-701: A proclamation in 1419, forbidding all those who had not borne arms at the Battle of Agincourt from assuming arms, except by inheritance or a grant from the crown. Beginning in the reign of Henry VIII of England, the English Kings of Arms were commanded to make visitations , in which they traveled about the country, recording arms borne under proper authority, and requiring those who bore arms without authority either to obtain authority for them, or cease their use. Arms borne improperly were to be taken down and defaced. The first such visitation began in 1530, and

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3410-578: A pseudotriangle, the partition gives 2 n − 2 {\displaystyle 2n-2} pseudotriangles and 3 n − 3 {\displaystyle 3n-3} bitangent lines. The convex hull of any pseudotriangle is a triangle. A non-planar triangle is a triangle not included in Euclidean space , roughly speaking a flat space. This means triangles may also be discovered in several spaces, as in hyperbolic space and spherical geometry . A triangle in hyperbolic space

3565-464: A reference triangle, the nearest points on the three sides serve as the vertices of the pedal triangle of that point. If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to

3720-401: A regular n -gon inscribed in a unit-radius circle, with side s and interior angle α , {\displaystyle \alpha ,} can also be expressed trigonometrically as: The area of a self-intersecting polygon can be defined in two different ways, giving different answers: Using the same convention for vertex coordinates as in the previous section, the coordinates of

3875-406: A right angle with it. The three perpendicular bisectors meet in a single point, the triangle's circumcenter ; this point is the center of the circumcircle , the circle passing through all three vertices. Thales' theorem implies that if the circumcenter is located on the side of the triangle, then the angle opposite that side is a right angle. If the circumcenter is located inside the triangle, then

4030-466: A shield divided azure and gules would be perfectly acceptable. A line of partition may be straight or it may be varied. The variations of partition lines can be wavy, indented, embattled, engrailed, nebuly , or made into myriad other forms; see Line (heraldry) . In the early days of heraldry, very simple bold rectilinear shapes were painted on shields. These could be easily recognized at a long distance and could be easily remembered. They therefore served

4185-432: A similar triangle: As discussed above, every triangle has a unique inscribed circle (incircle) that is interior to the triangle and tangent to all three sides. Every triangle has a unique Steiner inellipse which is interior to the triangle and tangent at the midpoints of the sides. Marden's theorem shows how to find the foci of this ellipse . This ellipse has the greatest area of any ellipse tangent to all three sides of

4340-399: A simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1. In every polygon with perimeter p and area A , the isoperimetric inequality p 2 > 4 π A {\displaystyle p^{2}>4\pi A} holds. For any two simple polygons of equal area,

4495-484: A simple polygon given by a sequence of line segments. This is called the point in polygon test. Heraldic Although the use of various devices to signify individuals and groups goes back to antiquity , both the form and use of such devices varied widely, as the concept of regular, hereditary designs, constituting the distinguishing feature of heraldry, did not develop until the High Middle Ages . It

4650-399: A simple polygon has a relationship to the ear , a vertex connected by two other vertices, the diagonal between which lies entirely within the polygon. The two ears theorem states that every simple polygon that is not itself a triangle has at least two ears. One way to identify locations of points in (or outside) a triangle is to place the triangle in an arbitrary location and orientation in

4805-491: A single point, the symmedian point of the triangle. The sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees. This fact is equivalent to Euclid's parallel postulate . This allows the determination of the measure of the third angle of any triangle, given the measure of two angles. An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary ) to an interior angle. The measure of an exterior angle of

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4960-413: A square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of

5115-477: A traditional shield under certain circumstances, and in Canadian heraldry the shield is now regularly granted. The whole surface of the escutcheon is termed the field , which may be plain, consisting of a single tincture, or divided into multiple sections of differing tinctures by various lines of partition; and any part of the field may be semé , or powdered with small charges. The edges and adjacent parts of

5270-407: A triangle are often constructed by proving that three symmetrically constructed points are collinear ; here Menelaus' theorem gives a useful general criterion. In this section, just a few of the most commonly encountered constructions are explained. A perpendicular bisector of a side of a triangle is a straight line passing through the midpoint of the side and being perpendicular to it, forming

5425-512: A triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the exterior angle theorem . The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees, and indeed, this is true for any convex polygon, no matter how many sides it has. Another relation between the internal angles and triangles creates a new concept of trigonometric functions . The primary trigonometric functions are sine and cosine , as well as

5580-404: A triangle of area at most equal to 2 T {\displaystyle 2T} . Equality holds only if the polygon is a parallelogram . The tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices. As mentioned above, every triangle has a unique circumcircle,

5735-504: A type of polygon (a skew polygon ), even when the chain does not lie in a single plane. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes. The word polygon derives from the Greek adjective πολύς ( polús ) 'much', 'many' and γωνία ( gōnía ) 'corner' or 'angle'. It has been suggested that γόνυ ( gónu ) 'knee' may be

5890-420: A unique line segment situated within a unique straight line , and any three points that do not all lie on the same straight line determine a unique triangle situated within a unique flat plane . More generally, four points in three-dimensional Euclidean space determine a tetrahedron . In non-Euclidean geometries , three "straight" segments (having zero curvature ) also determine a triangle, for instance,

6045-531: A window commemorating the knights who embarked on the Second Crusade in 1147, and was probably made soon after the event; but Montfaucon's illustration of the window before it was destroyed shows no heraldic design on any of the shields. In England, from the time of the Norman conquest, official documents had to be sealed. Beginning in the twelfth century, seals assumed a distinctly heraldic character;

6200-501: Is a cyclic hexagon with vertices given by the six intersections of the sides of a triangle with the three lines that are parallel to the sides and that pass through its symmedian point . In either its simple form or its self-intersecting form , the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle. Every convex polygon with area T {\displaystyle T} can be inscribed in

6355-430: Is a formula for finding the area of a triangle from the lengths of its sides a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} . Letting s = 1 2 ( a + b + c ) {\displaystyle s={\tfrac {1}{2}}(a+b+c)} be the semiperimeter , T = s ( s −

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6510-493: Is a solid whose boundary is covered by flat polygonals known as the faces, sharp corners known as the vertices, and line segments known as the edges. Polyhedra in some cases can be classified, judging from the shape of their faces. For example, when polyhedra have all equilateral triangles as their faces, they are known as deltahedra . Antiprisms have alternating triangles on their sides. Pyramids and bipyramids are polyhedra with polygonal bases and triangles for lateral faces;

6665-510: Is also credited with having originated the English crest of a lion statant (now statant-guardant). The origins of heraldry are sometimes associated with the Crusades , a series of military campaigns undertaken by Christian armies from 1096 to 1487, with the goal of reconquering Jerusalem and other former Byzantine territories captured by Muslim forces during the seventh century. While there

6820-701: Is any object or figure placed on a heraldic shield or on any other object of an armorial composition. Any object found in nature or technology may appear as a heraldic charge in armory. Charges can be animals, objects, or geometric shapes. Apart from the ordinaries, the most frequent charges are the cross – with its hundreds of variations – and the lion and eagle . Other common animals are bears , stags , wild boars , martlets , wolves and fish . Dragons , bats , unicorns , griffins , and other monsters appear as charges and as supporters . Animals are found in various stereotyped positions or attitudes . Quadrupeds can often be found rampant (standing on

6975-575: Is assumed throughout. Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are: In this section, the vertices of the polygon under consideration are taken to be ( x 0 , y 0 ) , ( x 1 , y 1 ) , … , ( x n − 1 , y n − 1 ) {\displaystyle (x_{0},y_{0}),(x_{1},y_{1}),\ldots ,(x_{n-1},y_{n-1})} in order. For convenience in some formulas,

7130-503: Is called barry , while a pattern of vertical (palewise) stripes is called paly . A pattern of diagonal stripes may be called bendy or bendy sinister , depending on the direction of the stripes. Other variations include chevrony , gyronny and chequy . Wave shaped stripes are termed undy . For further variations, these are sometimes combined to produce patterns of barry-bendy , paly-bendy , lozengy and fusilly . Semés, or patterns of repeated charges, are also considered variations of

7285-419: Is called a hyperbolic triangle , and it can be obtained by drawing on a negatively curved surface, such as a saddle surface . Likewise, a triangle in spherical geometry is called a spherical triangle , and it can be obtained by drawing on a positively curved surface such as a sphere . The triangles in both spaces have properties different from the triangles in Euclidean space. For example, as mentioned above,

7440-464: Is called a solid polygon . The interior of a solid polygon is its body , also known as a polygonal region or polygonal area . In contexts where one is concerned only with simple and solid polygons, a polygon may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polygons and other self-intersecting polygons . Some sources also consider closed polygonal chains in Euclidean space to be

7595-442: Is called an ermine. It consists of a white, or occasionally silver field, powdered with black figures known as ermine spots , representing the black tip of the animal's tail. Ermine was traditionally used to line the cloaks and caps of the nobility. The shape of the heraldic ermine spot has varied considerably over time, and nowadays is typically drawn as an arrowhead surmounted by three small dots, but older forms may be employed at

7750-439: Is commonly called the shoelace formula or surveyor's formula . The area A of a simple polygon can also be computed if the lengths of the sides, a 1 , a 2 , ..., a n and the exterior angles , θ 1 , θ 2 , ..., θ n are known, from: The formula was described by Lopshits in 1963. If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives

7905-438: Is given to the heraldic artist in depicting the heraldic tinctures; there is no fixed shade or hue to any of them. Whenever an object is depicted as it appears in nature, rather than in one or more of the heraldic tinctures, it is termed proper , or the colour of nature. This does not seem to have been done in the earliest heraldry, but examples are known from at least the seventeenth century. While there can be no objection to

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8060-451: Is large, this approaches one half. Or, each vertex inside the square mesh connects four edges (lines). The imaging system calls up the structure of polygons needed for the scene to be created from the database. This is transferred to active memory and finally, to the display system (screen, TV monitors etc.) so that the scene can be viewed. During this process, the imaging system renders polygons in correct perspective ready for transmission of

8215-462: Is more than two thousand years old, having been defined in Book One of Euclid's Elements . The names used for modern classification are either a direct transliteration of Euclid's Greek or their Latin translations. Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle , a triangle with two sides having

8370-507: Is no evidence that heraldic art originated in the course of the Crusades, there is no reason to doubt that the gathering of large armies, drawn from across Europe for a united cause, would have encouraged the adoption of armorial bearings as a means of identifying one's commanders in the field, or that it helped disseminate the principles of armory across Europe. At least two distinctive features of heraldry are generally accepted as products of

8525-423: Is no heraldic authority, and no law preventing anyone from assuming whatever arms they please, provided that they do not infringe upon the arms of another. Although heraldry originated from military necessity, it soon found itself at home in the pageantry of the medieval tournament . The opportunity for knights and lords to display their heraldic bearings in a competitive medium led to further refinements, such as

8680-452: Is normally left to the discretion of the heraldic artist, and many different shapes have prevailed during different periods of heraldic design, and in different parts of Europe. One shape alone is normally reserved for a specific purpose: the lozenge , a diamond-shaped escutcheon, was traditionally used to display the arms of women, on the grounds that shields, as implements of war, were inappropriate for this purpose. This distinction

8835-460: Is not adhered to quite as strictly. Arms which violate this rule are sometimes known as "puzzle arms", of which the most famous example is the arms of the Kingdom of Jerusalem , consisting of gold crosses on a silver field. The field of a shield, or less often a charge or crest, is sometimes made up of a pattern of colours, or variation . A pattern of horizontal (barwise) stripes, for example,

8990-399: Is not located on Euler's line. A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's centroid or geometric barycenter. The centroid of a rigid triangular object (cut out of a thin sheet of uniform density) is also its center of mass :

9145-424: Is often claimed that the use of helmets with face guards during this period made it difficult to recognize one's commanders in the field when large armies gathered together for extended periods, necessitating the development of heraldry as a symbolic language, but there is little support for this view. The perceived beauty and pageantry of heraldic designs allowed them to survive the gradual abandonment of armour on

9300-522: Is really no such thing as a stain in genuine heraldry, as well as the desire to create new and unique designs, the use of these colours for general purposes has become accepted in the twentieth and twenty-first centuries. Occasionally one meets with other colours, particularly in continental heraldry, although they are not generally regarded among the standard heraldic colours. Among these are cendrée , or ash-colour; brunâtre , or brown; bleu-céleste or bleu de ciel , sky blue; amaranth or columbine ,

9455-423: Is similar to vair in pale, but diagonal. When alternating rows are reversed as in counter-vair, and then displaced by half the width of one bell, it is termed vair in point , or wave-vair. A form peculiar to German heraldry is alternate vair , in which each vair bell is divided in half vertically, with half argent and half azure. All of these variations can also be depicted in the form known as potent , in which

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9610-452: Is sometimes found. Three additional furs are sometimes encountered in continental heraldry; in French and Italian heraldry one meets with plumeté or plumetty , in which the field appears to be covered with feathers, and papelonné , in which it is decorated with scales. In German heraldry one may encounter kursch , or vair bellies, depicted as brown and furry; all of these probably originated as variations of vair. Considerable latitude

9765-399: Is the matrix determinant . The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. Conversely, some triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality. The sum of two side lengths can equal the length of the third side only in

9920-407: Is the center of the triangle's incircle . The incircle is the circle that lies inside the triangle and touches all three sides. Its radius is called the inradius. There are three other important circles, the excircles ; they lie outside the triangle and touch one side, as well as the extensions of the other two. The centers of the incircles and excircles form an orthocentric system . The midpoints of

10075-410: Is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is acute. An angle bisector of a triangle is a straight line through a vertex that cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the incenter , which

10230-564: Is the use of a limited palette of colours and patterns, usually referred to as tinctures . These are divided into three categories, known as metals , colours , and furs . The metals are or and argent , representing gold and silver, respectively, although in practice they are usually depicted as yellow and white. Five colours are universally recognized: gules , or red; sable , or black; azure , or blue; vert , or green; and purpure , or purple; and most heraldic authorities also admit two additional colours, known as sanguine or murrey ,

10385-649: Is why engineering makes use of tetrahedral trusses . Triangulation means the partition of any planar object into a collection of triangles. For example, in polygon triangulation , a polygon is subdivided into multiple triangles that are attached edge-to-edge, with the property that their vertices coincide with the set of vertices of the polygon. In the case of a simple polygon with n {\displaystyle n} sides, there are n − 2 {\displaystyle n-2} triangles that are separated by n − 3 {\displaystyle n-3} diagonals. Triangulation of

10540-520: The Bayeux Tapestry , illustrating the Norman invasion of England in 1066, and probably commissioned about 1077, when the cathedral of Bayeux was rebuilt, depicts a number of shields of various shapes and designs, many of which are plain, while others are decorated with dragons, crosses, or other typically heraldic figures. Yet no individual is depicted twice bearing the same arms, nor are any of

10695-461: The Bolyai–Gerwien theorem asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon. The lengths of the sides of a polygon do not in general determine its area. However, if the polygon is simple and cyclic then the sides do determine the area. Of all n -gons with given side lengths, the one with the largest area is cyclic. Of all n -gons with

10850-497: The Cartesian plane , and to use Cartesian coordinates. While convenient for many purposes, this approach has the disadvantage of all points' coordinate values being dependent on the arbitrary placement in the plane. Two systems avoid that feature, so that the coordinates of a point are not affected by moving the triangle, rotating it, or reflecting it as in a mirror, any of which gives a congruent triangle, or even by rescaling it to

11005-635: The Giant's Causeway in Northern Ireland , or at the Devil's Postpile in California . In biology , the surface of the wax honeycomb made by bees is an array of hexagons , and the sides and base of each cell are also polygons. In computer graphics , a polygon is a primitive used in modelling and rendering. They are defined in a database, containing arrays of vertices (the coordinates of

11160-604: The Nebra sky disc , is also thought to serve as a heraldic precursor. Until the nineteenth century, it was common for heraldic writers to cite examples such as these, and metaphorical symbols such as the "Lion of Judah" or "Eagle of the Caesars", as evidence of the antiquity of heraldry itself; and to infer therefrom that the great figures of ancient history bore arms representing their noble status and descent. The Book of Saint Albans , compiled in 1486, declares that Christ himself

11315-410: The fess , the pale , the bend , the chevron , the saltire , and the pall . There is a separate class of charges called sub-ordinaries which are of a geometrical shape subordinate to the ordinary. According to Friar, they are distinguished by their order in blazon. The sub-ordinaries include the inescutcheon , the orle , the tressure, the double tressure, the bordure , the chief , the canton ,

11470-456: The geometrical vertices , as well as other attributes of the polygon, such as color, shading and texture), connectivity information, and materials . Any surface is modelled as a tessellation called polygon mesh . If a square mesh has n + 1 points (vertices) per side, there are n squared squares in the mesh, or 2 n squared triangles since there are two triangles in a square. There are ( n + 1) / 2( n ) vertices per triangle. Where n

11625-604: The griffin can also be found. In the Bible , the Book of Numbers refers to the standards and ensigns of the children of Israel , who were commanded to gather beneath these emblems and declare their pedigrees. The Greek and Latin writers frequently describe the shields and symbols of various heroes, and units of the Roman army were sometimes identified by distinctive markings on their shields. At least one pre-historic European object,

11780-404: The herald , originally a type of messenger employed by noblemen, assumed the responsibility of learning and knowing the rank, pedigree, and heraldic devices of various knights and lords, as well as the rules governing the design and description, or blazoning of arms, and the precedence of their bearers. As early as the late thirteenth century, certain heralds in the employ of monarchs were given

11935-459: The label , and flaunches . Ordinaries may appear in parallel series, in which case blazons in English give them different names such as pallets, bars, bendlets, and chevronels. French blazon makes no such distinction between these diminutives and the ordinaries when borne singly. Unless otherwise specified an ordinary is drawn with straight lines, but each may be indented, embattled, wavy, engrailed, or otherwise have their lines varied. A charge

12090-488: The regular star pentagon is also known as the pentagram . To construct the name of a polygon with more than 20 and fewer than 100 edges, combine the prefixes as follows. The "kai" term applies to 13-gons and higher and was used by Kepler , and advocated by John H. Conway for clarity of concatenated prefix numbers in the naming of quasiregular polyhedra , though not all sources use it. Polygons have been known since ancient times. The regular polygons were known to

12245-485: The simplicial polytopes . Each triangle has many special points inside it, on its edges, or otherwise associated with it. They are constructed by finding three lines associated symmetrically with the three sides (or vertices) and then proving that the three lines meet in a single point. An important tool for proving the existence of these points is Ceva's theorem , which gives a criterion for determining when three such lines are concurrent . Similarly, lines associated with

12400-420: The altitude can be calculated using trigonometry, ⁠ h = a sin ⁡ ( γ ) {\displaystyle h=a\sin(\gamma )} ⁠ , so the area of the triangle is: T = 1 2 a b sin ⁡ γ . {\displaystyle T={\tfrac {1}{2}}ab\sin \gamma .} Heron's formula , named after Heron of Alexandria ,

12555-539: The ancient Greeks, with the pentagram , a non-convex regular polygon ( star polygon ), appearing as early as the 7th century B.C. on a krater by Aristophanes , found at Caere and now in the Capitoline Museum . The first known systematic study of non-convex polygons in general was made by Thomas Bradwardine in the 14th century. In 1952, Geoffrey Colin Shephard generalized the idea of polygons to

12710-827: The angles of a triangle can also be stated using trigonometric functions. For example, a triangle with angles α {\displaystyle \alpha } , β {\displaystyle \beta } , and γ {\displaystyle \gamma } exists if and only if cos 2 ⁡ α + cos 2 ⁡ β + cos 2 ⁡ γ + 2 cos ⁡ ( α ) cos ⁡ ( β ) cos ⁡ ( γ ) = 1. {\displaystyle \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma +2\cos(\alpha )\cos(\beta )\cos(\gamma )=1.} Two triangles are said to be similar , if every angle of one triangle has

12865-417: The angles of a triangle on a sphere is 180 ∘ × ( 1 + 4 f ) {\displaystyle 180^{\circ }\times (1+4f)} , where f {\displaystyle f} is the fraction of the sphere's area enclosed by the triangle. In more general spaces, there are comparison theorems relating the properties of a triangle in the space to properties of

13020-403: The antiquity of heraldry. The development of the modern heraldic language cannot be attributed to a single individual, time, or place. Although certain designs that are now considered heraldic were evidently in use during the eleventh century, most accounts and depictions of shields up to the beginning of the twelfth century contain little or no evidence of their heraldic character. For example,

13175-437: The area of an arbitrary triangle. One of the oldest and simplest is to take half the product of the length of one side ⁠ b {\displaystyle b} ⁠ (the base) times the corresponding altitude ⁠ h {\displaystyle h} ⁠ : T = 1 2 b h . {\displaystyle T={\tfrac {1}{2}}bh.} This formula can be proven by cutting up

13330-409: The arms of a married couple, that the wife is an heraldic heiress (i.e., she inherits a coat of arms because she has no brothers). In continental Europe an inescutcheon (sometimes called a "heart shield") usually carries the ancestral arms of a monarch or noble whose domains are represented by the quarters of the main shield. In German heraldry , animate charges in combined coats usually turn to face

13485-404: The artist's discretion. When the field is sable and the ermine spots argent, the same pattern is termed ermines ; when the field is or rather than argent, the fur is termed erminois ; and when the field is sable and the ermine spots or , it is termed pean . Vair represents the winter coat of the red squirrel , which is blue-grey on top and white underneath. To form the linings of cloaks,

13640-612: The authority of the Earl Marshal ; but all of the arms granted by the college are granted by the authority of the crown. In Scotland Court of the Lord Lyon King of Arms oversees the heraldry, and holds court sessions which are an official part of Scotland's court system. Similar bodies regulate the granting of arms in other monarchies and several members of the Commonwealth of Nations , but in most other countries there

13795-423: The base of length a {\displaystyle a} is equal to a {\displaystyle a} . The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is 2 2 / 3 {\displaystyle 2{\sqrt {2}}/3} . Both of these extreme cases occur for the isosceles right triangle. The Lemoine hexagon

13950-435: The base. The other points include dexter chief , center chief , and sinister chief , running along the upper part of the shield from left to right, above the honour point; dexter flank and sinister flank , on the sides approximately level with fess point; and dexter base , middle base , and sinister base along the lower part of the shield, below the nombril point. One of the most distinctive qualities of heraldry

14105-646: The battlefield during the seventeenth century. Heraldry has been described poetically as "the handmaid of history", "the shorthand of history", and "the floral border in the garden of history". In modern times, individuals, public and private organizations, corporations, cities, towns, regions, and other entities use heraldry and its conventions to symbolize their heritage, achievements, and aspirations. Various symbols have been used to represent individuals or groups for thousands of years. The earliest representations of distinct persons and regions in Egyptian art show

14260-420: The case of a degenerate triangle , one with collinear vertices. Unlike a rectangle, which may collapse into a parallelogram from pressure to one of its points, triangles are sturdy because specifying the lengths of all three sides determines the angles. Therefore, a triangle will not change shape unless its sides are bent or extended or broken or if its joints break; in essence, each of the three sides supports

14415-406: The center of the nine-point circle (red), the centroid (orange), and the circumcenter (green) all lie on a single line, known as Euler's line (red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter. Generally, the incircle's center

14570-416: The centre of the composition. In English the word "crest" is commonly (but erroneously) used to refer to an entire heraldic achievement of armorial bearings. The technical use of the heraldic term crest refers to just one component of a complete achievement. The crest rests on top of a helmet which itself rests on the most important part of the achievement: the shield. The modern crest has grown out of

14725-453: The centroid of a solid simple polygon are In these formulas, the signed value of area A {\displaystyle A} must be used. For triangles ( n = 3 ), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for n > 3 . The centroid of the vertex set of a polygon with n vertices has the coordinates The idea of a polygon has been generalized in various ways. Some of

14880-426: The complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons . Polygons appear in rock formations, most commonly as the flat facets of crystals , where the angles between the sides depend on the type of mineral from which the crystal is made. Regular hexagons can occur when the cooling of lava forms areas of tightly packed columns of basalt , which may be seen at

15035-401: The crusaders: the surcoat , an outer garment worn over the armor to protect the wearer from the heat of the sun, was often decorated with the same devices that appeared on a knight's shield. It is from this garment that the phrase "coat of arms" is derived. Also the lambrequin, or mantling, that depends from the helmet and frames the shield in modern heraldry, began as a practical covering for

15190-532: The descendants of the various persons depicted known to have borne devices resembling those in the tapestry. Similarly, an account of the French knights at the court of the Byzantine emperor Alexius I at the beginning of the twelfth century describes their shields of polished metal, devoid of heraldic design. A Spanish manuscript from 1109 describes both plain and decorated shields, none of which appears to have been heraldic. The Abbey of St. Denis contained

15345-405: The development of "landscape heraldry", incorporating realistic depictions of landscapes, during the latter part of the eighteenth and early part of the nineteenth century. These fell out of fashion during the mid-nineteenth century, when a renewed interest in the history of armory led to the re-evaluation of earlier designs, and a new appreciation for the medieval origins of the art. In particular,

15500-471: The development of elaborate tournament helms, and further popularized the art of heraldry throughout Europe. Prominent burghers and corporations, including many cities and towns, assumed or obtained grants of arms, with only nominal military associations. Heraldic devices were depicted in various contexts, such as religious and funerary art, and in using a wide variety of media, including stonework, carved wood, enamel , stained glass , and embroidery . As

15655-446: The dexter is on the left side, and the sinister on the right. The placement of various charges may also refer to a number of specific points, nine in number according to some authorities, but eleven according to others. The three most important are fess point , located in the visual center of the shield; the honour point , located midway between fess point and the chief; and the nombril point , located midway between fess point and

15810-443: The earliest known examples of armory as it subsequently came to be practiced can be seen on the tomb of Geoffrey Plantagenet, Count of Anjou , who died in 1151. An enamel, probably commissioned by Geoffrey's widow between 1155 and 1160, depicts him carrying a blue shield decorated with six golden lions rampant. He wears a blue helmet adorned with another lion, and his cloak is lined in vair. A medieval chronicle states that Geoffrey

15965-436: The escutcheon are used to identify the placement of various heraldic charges; the upper edge, and the corresponding upper third of the shield, are referred to as the chief; the lower part is the base. The sides of the shield are known as the dexter and sinister flanks, although these terms are based on the point of view of the bearer of the shield, who would be standing behind it; to the observer, and in all heraldic illustration,

16120-399: The field. The Rule of tincture applies to all semés and variations of the field. The field of a shield in heraldry can be divided into more than one tincture , as can the various heraldic charges . Many coats of arms consist simply of a division of the field into two contrasting tinctures. These are considered divisions of a shield, so the rule of tincture can be ignored. For example,

16275-475: The fur is termed gros vair or beffroi ; if of six or more, it is menu-vair , or miniver. A common variation is counter-vair , in which alternating rows are reversed, so that the bases of the vair bells of each tincture are joined to those of the same tincture in the row above or below. When the rows are arranged so that the bells of each tincture form vertical columns, it is termed vair in pale ; in continental heraldry one may encounter vair in bend , which

16430-402: The helmet and the back of the neck during the Crusades, serving much the same function as the surcoat. Its slashed or scalloped edge, today rendered as billowing flourishes, is thought to have originated from hard wearing in the field, or as a means of deadening a sword blow and perhaps entangling the attacker's weapon. The spread of armorial bearings across Europe gave rise to a new occupation:

16585-408: The internal angles of a triangle in Euclidean space always add up to 180°. However, the sum of the internal angles of a hyperbolic triangle is less than 180°, and for any spherical triangle, the sum is more than 180°. In particular, it is possible to draw a triangle on a sphere such that the measure of each of its internal angles equals 90°, adding up to a total of 270°. By Girard's theorem , the sum of

16740-585: The isosceles triangles may be found in the shape of gables and pediments , and the equilateral triangle can be found in the yield sign. The faces of the Great Pyramid of Giza are sometimes considered to be equilateral, but more accurate measurements show they are isosceles instead. Other appearances are in heraldic symbols as in the flag of Saint Lucia and flag of the Philippines . Triangles also appear in three-dimensional objects. A polyhedron

16895-567: The last was carried out in 1700, although no new commissions to carry out visitations were made after the accession of William III in 1689. There is little evidence that Scottish heralds ever went on visitations. In 1484, during the reign of Richard III , the various heralds employed by the crown were incorporated into England's College of Arms , through which all new grants of arms would eventually be issued. The college currently consists of three Kings of Arms, assisted by six Heralds, and four Pursuivants , or junior officers of arms, all under

17050-409: The left hind foot). Another frequent position is passant , or walking, like the lions of the coat of arms of England . Eagles are almost always shown with their wings spread, or displayed. A pair of wings conjoined is called a vol . In English heraldry the crescent , mullet , martlet , annulet , fleur-de-lis , and rose may be added to a shield to distinguish cadet branches of a family from

17205-414: The lengths of their sides. Relations between angles and side lengths are a major focus of trigonometry . In particular, the sine, cosine, and tangent functions relate side lengths and angles in right triangles . A triangle is a figure consisting of three line segments, each of whose endpoints are connected. This forms a polygon with three sides and three angles. The terminology for categorizing triangles

17360-399: The main purpose of heraldry: identification. As more complicated shields came into use, these bold shapes were set apart in a separate class as the "honourable ordinaries". They act as charges and are always written first in blazon . Unless otherwise specified they extend to the edges of the field. Though ordinaries are not easily defined, they are generally described as including the cross ,

17515-833: The more important include: The word polygon comes from Late Latin polygōnum (a noun), from Greek πολύγωνον ( polygōnon/polugōnon ), noun use of neuter of πολύγωνος ( polygōnos/polugōnos , the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek -derived numerical prefix with the suffix -gon , e.g. pentagon , dodecagon . The triangle , quadrilateral and nonagon are exceptions. Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon. Exceptions exist for side counts that are easily expressed in verbal form (e.g. 20 and 30), or are used by non-mathematicians. Some special polygons also have their own names; for example

17670-430: The name implies, the usual number of divisions is four, but the principle has been extended to very large numbers of "quarters". Quarters are numbered from the dexter chief (the corner nearest to the right shoulder of a man standing behind the shield), proceeding across the top row, and then across the next row and so on. When three coats are quartered, the first is repeated as the fourth; when only two coats are quartered,

17825-685: The next, representing a particular person or line of descent. The medieval heralds also devised arms for various knights and lords from history and literature. Notable examples include the toads attributed to Pharamond , the cross and martlets of Edward the Confessor , and the various arms attributed to the Nine Worthies and the Knights of the Round Table . These too are readily dismissed as fanciful inventions, rather than evidence of

17980-552: The notation ( x n , y n ) = ( x 0 , y 0 ) will also be used. If the polygon is non-self-intersecting (that is, simple ), the signed area is or, using determinants where Q i , j {\displaystyle Q_{i,j}} is the squared distance between ( x i , y i ) {\displaystyle (x_{i},y_{i})} and ( x j , y j ) . {\displaystyle (x_{j},y_{j}).} The signed area depends on

18135-406: The object can be balanced on its centroid in a uniform gravitational field. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice the distance between the centroid and the midpoint of the opposite side. If one reflects a median in the angle bisector that passes through the same vertex, one obtains a symmedian . The three symmedians intersect in

18290-601: The occasional depiction of objects in this manner, the overuse of charges in their natural colours is often cited as indicative of bad heraldic practice. The practice of landscape heraldry, which flourished in the latter part of the eighteenth and early part of the nineteenth century, made extensive use of non-heraldic colours. One of the most important conventions of heraldry is the so-called " rule of tincture ". To provide for contrast and visibility, metals should never be placed on metals, and colours should never be placed on colours. This rule does not apply to charges which cross

18445-401: The older, undulating pattern, now known as vair ondé or vair ancien , the bells of each tincture are curved and joined at the base. There is no fixed rule as to whether the argent bells should be at the top or the bottom of each row. At one time vair commonly came in three sizes, and this distinction is sometimes encountered in continental heraldry; if the field contains fewer than four rows,

18600-423: The ordering of the vertices and of the orientation of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive x -axis to the positive y -axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. In either case, the area formula is correct in absolute value . This

18755-427: The origin of gon . Polygons are primarily classified by the number of sides. Polygons may be characterized by their convexity or type of non-convexity: The property of regularity may be defined in other ways: a polygon is regular if and only if it is both isogonal and isotoxal, or equivalently it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon . Euclidean geometry

18910-407: The other functions. They can be defined as the ratio between any two sides of a right triangle . In a scalene triangle, the trigonometric functions can be used to find the unknown measure of either a side or an internal angle; methods for doing so use the law of sines and the law of cosines . Any three angles that add to 180° can be the internal angles of a triangle. Infinitely many triangles have

19065-411: The other two. A rectangle, in contrast, is more dependent on the strength of its joints in a structural sense. Triangles are strong in terms of rigidity, but while packed in a tessellating arrangement triangles are not as strong as hexagons under compression (hence the prevalence of hexagonal forms in nature ). Tessellated triangles still maintain superior strength for cantilevering , however, which

19220-410: The pelts were sewn together, forming an undulating, bell-shaped pattern, with interlocking light and dark rows. The heraldic fur is depicted with interlocking rows of argent and azure, although the shape of the pelts, usually referred to as "vair bells", is usually left to the artist's discretion. In the modern form, the bells are depicted with straight lines and sharp angles, and meet only at points; in

19375-413: The polygon's vertices or corners . An n -gon is a polygon with n sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that

19530-426: The processed data to the display system. Although polygons are two-dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation. In computer graphics and computational geometry , it is often necessary to determine whether a given point P = ( x 0 , y 0 ) {\displaystyle P=(x_{0},y_{0})} lies inside

19685-431: The reference triangle. The intouch triangle of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle. The extouch triangle of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides (not extended). Every acute triangle has three inscribed squares (squares in its interior such that all four of

19840-462: The right from the viewpoint of the bearer of the arms and "sinister" (from Latin sinistra , "left") means to the bearer's left. The dexter side is considered the side of greatest honour (see also dexter and sinister ). A more versatile method is quartering , division of the field by both vertical and horizontal lines. This practice originated in Spain ( Castile and León ) after the 13th century. As

19995-418: The rise of firearms rendered the mounted knight increasingly irrelevant during the sixteenth and seventeenth centuries, and the tournament faded into history, the military character of heraldry gave way to its use as a decorative art. Freed from the limitations of actual shields and the need for arms to be easily distinguished in combat, heraldic artists designed increasingly elaborate achievements, culminating in

20150-509: The same angles, since specifying the angles of a triangle does not determine its size. (A degenerate triangle , whose vertices are collinear , has internal angles of 0° and 180°; whether such a shape counts as a triangle is a matter of convention.) The conditions for three angles α {\displaystyle \alpha } , β {\displaystyle \beta } , and γ {\displaystyle \gamma } , each of them between 0° and 180°, to be

20305-771: The same base and oriented area has its apex (the third vertex) on a line parallel to the base, and their common area is half of that of a parallelogram with the same base whose opposite side lies on the parallel line. This affine approach was developed in Book 1 of Euclid's Elements . Given affine coordinates (such as Cartesian coordinates ) ⁠ ( x A , y A ) {\displaystyle (x_{A},y_{A})} ⁠ , ⁠ ( x B , y B ) {\displaystyle (x_{B},y_{B})} ⁠ , ⁠ ( x C , y C ) {\displaystyle (x_{C},y_{C})} ⁠ for

20460-511: The same length is an isosceles triangle , and a triangle with three different-length sides is a scalene triangle . A triangle in which one of the angles is a right angle is a right triangle , a triangle in which all of its angles are less than that angle is an acute triangle , and a triangle in which one of it angles is greater than that angle is an obtuse triangle . These definitions date back at least to Euclid . All types of triangles are commonly found in real life. In man-made construction,

20615-410: The same length. This is a total of six equalities, but three are often sufficient to prove congruence. Some individually necessary and sufficient conditions for a pair of triangles to be congruent are: In the Euclidean plane, area is defined by comparison with a square of side length ⁠ 1 {\displaystyle 1} ⁠ , which has area 1. There are several ways to calculate

20770-590: The same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity. Some basic theorems about similar triangles are: Two triangles that are congruent have exactly the same size and shape. All pairs of congruent triangles are also similar, but not all pairs of similar triangles are congruent. Given two congruent triangles, all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have

20925-433: The second is also repeated as the third. The quarters of a personal coat of arms correspond to the ancestors from whom the bearer has inherited arms, normally in the same sequence as if the pedigree were laid out with the father's father's ... father (to as many generations as necessary) on the extreme left and the mother's mother's...mother on the extreme right. A few lineages have accumulated hundreds of quarters, though such

21080-426: The senior line. These cadency marks are usually shown smaller than normal charges, but it still does not follow that a shield containing such a charge belongs to a cadet branch. All of these charges occur frequently in basic undifferenced coats of arms. To marshal two or more coats of arms is to combine them in one shield, to express inheritance, claims to property, or the occupation of an office. This can be done in

21235-585: The shape of the vair bell is replaced by a T -shaped figure, known as a potent from its resemblance to a crutch. Although it is really just a variation of vair, it is frequently treated as a separate fur. When the same patterns are composed of tinctures other than argent and azure, they are termed vairé or vairy of those tinctures, rather than vair ; potenté of other colours may also be found. Usually vairé will consist of one metal and one colour, but ermine or one of its variations may also be used, and vairé of four tinctures, usually two metals and two colours,

21390-436: The shield. Some arms, particularly those of the nobility, are further embellished with supporters, heraldic figures standing alongside or behind the shield; often these stand on a compartment , typically a mound of earth and grass, on which other badges , symbols, or heraldic banners may be displayed. The most elaborate achievements sometimes display the entire coat of arms beneath a pavilion, an embellished tent or canopy of

21545-408: The shields. These in turn came to be decorated with fan-shaped or sculptural crests, often incorporating elements from the shield of arms; as well as a wreath or torse , or sometimes a coronet , from which depended the lambrequin or mantling . To these elements, modern heraldry often adds a motto displayed on a ribbon, typically below the shield. The helmet is borne of right, and forms no part of

21700-487: The three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle . The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter . The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point ) and the three excircles . The orthocenter (blue point),

21855-548: The title "King of Heralds", which eventually became " King of Arms ." In the earliest period, arms were assumed by their bearers without any need for heraldic authority. However, by the middle of the fourteenth century, the principle that only a single individual was entitled to bear a particular coat of arms was generally accepted, and disputes over the ownership of arms seems to have led to gradual establishment of heraldic authorities to regulate their use. The earliest known work of heraldic jurisprudence , De Insigniis et Armis ,

22010-491: The triangle and an identical copy into pieces and rearranging the pieces into the shape of a rectangle of base ⁠ b {\displaystyle b} ⁠ and height ⁠ h {\displaystyle h} ⁠ . If two sides ⁠ a {\displaystyle a} ⁠ and ⁠ b {\displaystyle b} ⁠ and their included angle γ {\displaystyle \gamma } are known, then

22165-400: The triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse. An altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. The length of the altitude

22320-439: The triangle's longest side. Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has a side of length q a {\displaystyle q_{a}} and the triangle has a side of length a {\displaystyle a} , part of which side coincides with a side of the square, then q a {\displaystyle q_{a}} ,

22475-455: The triangle's three vertices, its centroid, and its circumcenter. Of all triangles contained in a given convex polygon , one with maximal area can be found in linear time; its vertices may be chosen as three of the vertices of the given polygon. A circular triangle is a triangle with circular arc edges. The edges of a circular triangle may be either convex (bending outward) or concave (bending inward). The intersection of three disks forms

22630-1278: The triangle. The Mandart inellipse of a triangle is the ellipse inscribed within the triangle tangent to its sides at the contact points of its excircles. For any ellipse inscribed in a triangle A B C {\displaystyle ABC} , let the foci be P {\displaystyle P} and Q {\displaystyle Q} , then: P A ¯ ⋅ Q A ¯ C A ¯ ⋅ A B ¯ + P B ¯ ⋅ Q B ¯ A B ¯ ⋅ B C ¯ + P C ¯ ⋅ Q C ¯ B C ¯ ⋅ C A ¯ = 1. {\displaystyle {\frac {{\overline {PA}}\cdot {\overline {QA}}}{{\overline {CA}}\cdot {\overline {AB}}}}+{\frac {{\overline {PB}}\cdot {\overline {QB}}}{{\overline {AB}}\cdot {\overline {BC}}}}+{\frac {{\overline {PC}}\cdot {\overline {QC}}}{{\overline {BC}}\cdot {\overline {CA}}}}=1.} From an interior point in

22785-418: The triangles are isosceles whenever they are right pyramids and bipyramids. The Kleetope of a polyhedron is a new polyhedron made by replacing each face of the original with a pyramid, and so the faces of a Kleetope will be triangles. More generally, triangles can be found in higher dimensions, as in the generalized notion of triangles known as the simplex , and the polytopes with triangular facets known as

22940-434: The type associated with the medieval tournament, though this is only very rarely found in English or Scots achievements. The primary element of a heraldic achievement is the shield, or escutcheon, upon which the coat of arms is depicted. All of the other elements of an achievement are designed to decorate and complement these arms, but only the shield of arms is required. The shape of the shield, like many other details,

23095-421: The use of standards topped with the images or symbols of various gods, and the names of kings appear upon emblems known as serekhs , representing the king's palace, and usually topped with a falcon representing the god Horus , of whom the king was regarded as the earthly incarnation. Similar emblems and devices are found in ancient Mesopotamian art of the same period, and the precursors of heraldic beasts such as

23250-1702: The vertices of a triangle, its relative oriented area can be calculated using the shoelace formula , T = 1 2 | x A x B x C y A y B y C 1 1 1 | = 1 2 | x A x B y A y B | + 1 2 | x B x C y B y C | + 1 2 | x C x A y C y A | = 1 2 ( x A y B − x B y A + x B y C − x C y B + x C y A − x A y C ) , {\displaystyle {\begin{aligned}T&={\tfrac {1}{2}}{\begin{vmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{vmatrix}}={\tfrac {1}{2}}{\begin{vmatrix}x_{A}&x_{B}\\y_{A}&y_{B}\end{vmatrix}}+{\tfrac {1}{2}}{\begin{vmatrix}x_{B}&x_{C}\\y_{B}&y_{C}\end{vmatrix}}+{\tfrac {1}{2}}{\begin{vmatrix}x_{C}&x_{A}\\y_{C}&y_{A}\end{vmatrix}}\\&={\tfrac {1}{2}}(x_{A}y_{B}-x_{B}y_{A}+x_{B}y_{C}-x_{C}y_{B}+x_{C}y_{A}-x_{A}y_{C}),\end{aligned}}} where | ⋅ | {\displaystyle |\cdot |}

23405-414: Was a gentleman of coat armour. These claims are now regarded as the fantasy of medieval heralds, as there is no evidence of a distinctive symbolic language akin to that of heraldry during this early period; nor do many of the shields described in antiquity bear a close resemblance to those of medieval heraldry; nor is there any evidence that specific symbols or designs were passed down from one generation to

23560-523: Was given a shield of this description when he was knighted by his father-in-law, Henry I , in 1128; but this account probably dates to about 1175. The earlier heraldic writers attributed the lions of England to William the Conqueror , but the earliest evidence of the association of lions with the English crown is a seal bearing two lions passant, used by the future King John during the lifetime of his father, Henry II , who died in 1189. Since Henry

23715-567: Was not always strictly adhered to, and a general exception was usually made for sovereigns, whose arms represented an entire nation. Sometimes an oval shield, or cartouche, was substituted for the lozenge; this shape was also widely used for the arms of clerics in French, Spanish, and Italian heraldry, although it was never reserved for their use. In recent years, the use of the cartouche for women's arms has become general in Scottish heraldry, while both Scottish and Irish authorities have permitted

23870-555: Was the son of Geoffrey Plantagenet, it seems reasonable to suppose that the adoption of lions as an heraldic emblem by Henry or his sons might have been inspired by Geoffrey's shield. John's elder brother, Richard the Lionheart , who succeeded his father on the throne, is believed to have been the first to have borne the arms of three lions passant-guardant, still the arms of England, having earlier used two lions rampant combatant, which arms may also have belonged to his father. Richard

24025-474: Was written about 1350 by Bartolus de Saxoferrato , a professor of law at the University of Padua . The most celebrated armorial dispute in English heraldry is that of Scrope v Grosvenor (1390), in which two different men claimed the right to bear azure, a bend or . The continued proliferation of arms, and the number of disputes arising from different men assuming the same arms, led Henry V to issue

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