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93-459: Familiar examples of inscribed figures include circles inscribed in triangles or regular polygons , and triangles or regular polygons inscribed in circles. A circle inscribed in any polygon is called its incircle , in which case the polygon is said to be a tangential polygon . A polygon inscribed in a circle is said to be a cyclic polygon , and the circle is said to be its circumscribed circle or circumcircle . The inradius or filling radius of

186-415: 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

279-404: 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

372-441: 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

465-454: 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

558-398: 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

651-521: 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

744-499: A corresponding triangle in a model space like hyperbolic or elliptic space. For example, a CAT(k) space is characterized by such comparisons. Euclid%27s Elements The Elements ( ‹See Tfd› Greek : Στοιχεῖα Stoikheîa ) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid c. 300 BC. It is a collection of definitions, postulates , propositions ( theorems and constructions ), and mathematical proofs of

837-637: A geometry which assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different versions of the fifth postulate ( elliptic geometry ). If one takes the fifth postulate as a given, the result is Euclidean geometry . • "To draw a straight line from any point to any point." • "To describe a circle with any center and distance." Euclid, Elements , Book I, Postulates 1 & 3. Euclid's axiomatic approach and constructive methods were widely influential. Many of Euclid's propositions were constructive, demonstrating

930-450: A given outer figure is the radius of the inscribed circle or sphere, if it exists. The definition given above assumes that the objects concerned are embedded in two- or three- dimensional Euclidean space , but can easily be generalized to higher dimensions and other metric spaces . For an alternative usage of the term "inscribed", see the inscribed square problem , in which a square is considered to be inscribed in another figure (even

1023-466: A glimpse of an otherworldly system of perfection and certainty. Abraham Lincoln kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield , went home to my father's house, and stayed there till I could give any proposition in

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1116-540: A line and circle. It also appears that, for him to use a figure in one of his proofs, he needs to construct it in an earlier proposition. For example, he proves the Pythagorean theorem by first inscribing a square on the sides of a right triangle, but only after constructing a square on a given line one proposition earlier. As was common in ancient mathematical texts, when a proposition needed proof in several different cases, Euclid often proved only one of them (often

1209-457: A new concept of trigonometric functions . The primary trigonometric functions are sine and cosine , as well as 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

1302-414: A non-convex one) if all four of its vertices are on that figure. Triangle 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

1395-409: A pair of adjacent edges; the sum of angles of a triangle always equals 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

1488-580: 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

1581-464: 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

1674-409: 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

1767-434: 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

1860-400: 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

1953-494: 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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2046-413: 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

2139-533: A thousand different editions. Theon's Greek edition was recovered and published in 1533 based on Paris gr. 2343 and Venetus Marcianus 301. In 1570, John Dee provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by Henry Billingsley . Copies of the Greek text still exist, some of which can be found in the Vatican Library and

2232-408: 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

2325-400: 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

2418-406: 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,

2511-587: A triangle, for instance, 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

2604-502: 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

2697-431: 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 −

2790-495: 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;

2883-420: 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,

Inscribed figure - Misplaced Pages Continue

2976-464: 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

3069-401: 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 :

3162-403: Is recognized as typically classical. It has six different parts: First is the 'enunciation', which states the result in general terms (i.e., the statement of the proposition). Then comes the 'setting-out', which gives the figure and denotes particular geometrical objects by letters. Next comes the 'definition' or 'specification', which restates the enunciation in terms of the particular figure. Then

3255-504: Is still an active area of research. Campanus of Novara relied heavily on these Arabic translations to create his edition (sometime before 1260) which ultimately came to dominate Latin editions until the availability of Greek manuscripts in the 16th century. There are more than 100 pre-1482 Campanus manuscripts still available today. The first printed edition appeared in 1482 (based on Campanus's translation), and since then it has been translated into many languages and published in about

3348-402: 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

3441-408: 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

3534-412: 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

3627-496: Is unique conic that passes through 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

3720-651: 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

3813-718: The Bodleian Library in Oxford. The manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text (copies of which are no longer available). Ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of

Inscribed figure - Misplaced Pages Continue

3906-498: 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

3999-645: The Elements from the Byzantines around 760; this version was translated into Arabic under Harun al-Rashid ( c. 800). The Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, when the English monk Adelard of Bath translated it into Latin from an Arabic translation. A relatively recent discovery

4092-547: The Elements is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Much of the material is not original to him, although many of the proofs are his. However, Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the Elements , encouraged its use as a textbook for about 2,000 years. The Elements still influences modern geometry books. Furthermore, its logical, axiomatic approach and rigorous proofs remain

4185-452: The Elements , and applied their knowledge of it to their work. Mathematicians and philosophers, such as Thomas Hobbes , Baruch Spinoza , Alfred North Whitehead , and Bertrand Russell , have attempted to create their own foundational "Elements" for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced. The austere beauty of Euclidean geometry has been seen by many in western culture as

4278-705: The Elements : "Euclid, who put together the Elements , collecting many of Eudoxus ' theorems, perfecting many of Theaetetus ', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors". Pythagoras ( c. 570–495 BC) was probably the source for most of books I and II, Hippocrates of Chios ( c. 470–410 BC, not the better known Hippocrates of Kos ) for book III, and Eudoxus of Cnidus ( c. 408–355 BC) for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians. The Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated

4371-471: The Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions. Papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although Euclid was known to Cicero , for instance, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received

4464-487: 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

4557-492: The 'construction' or 'machinery' follows. Here, the original figure is extended to forward the proof. Then, the 'proof' itself follows. Finally, the 'conclusion' connects the proof to the enunciation by stating the specific conclusions drawn in the proof, in the general terms of the enunciation. No indication is given of the method of reasoning that led to the result, although the Data does provide instruction about how to approach

4650-420: 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 ,

4743-828: 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

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4836-419: 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

4929-400: The angles sum to less than two right angles. This postulate plagued mathematicians for centuries due to its apparent complexity compared with the other four postulates. Many attempts were made to prove the fifth postulate based on the other four, but they never succeeded. Eventually in 1829, mathematician Nikolai Lobachevsky published a description of acute geometry (or hyperbolic geometry ),

5022-438: 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

5115-425: 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

5208-422: 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

5301-407: 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

5394-439: The chief result being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being 10 3 ( 5 − 5 ) = 5 + 5 6 . {\displaystyle {\sqrt {\frac {10}{3(5-{\sqrt {5}})}}}={\sqrt {\frac {5+{\sqrt {5}}}{6}}}.} The spurious Book XV

5487-432: The cornerstone of mathematics. One of the most notable influences of Euclid on modern mathematics is the discussion of the parallel postulate . In Book I, Euclid lists five postulates, the fifth of which stipulates If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles , then the two lines, if extended indefinitely, meet on that side on which

5580-606: The errors include Hilbert's geometry axioms and Tarski's . In 2018, Michael Beeson et al. used computer proof assistants to create a new set of axioms similar to Euclid's and generate proofs that were valid with those axioms. Beeson et al. checked only Book I and found these errors: missing axioms, superfluous axioms, gaps in logic (such as failing to prove points were colinear), missing theorems (such as an angle cannot be less than itself), and outright bad proofs. The bad proofs were in Book I, Proof 7 and Book I, Proposition 9. It

5673-403: The existence of some figure by detailing the steps he used to construct the object using a compass and straightedge . His constructive approach appears even in his geometry's postulates, as the first and third postulates stating the existence of a line and circle are constructive. Instead of stating that lines and circles exist per his prior definitions, he states that it is possible to 'construct'

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5766-457: The figure used as an example to illustrate one given configuration. Euclid's Elements contains errors. Some of the foundational theorems are proved using axioms that Euclid did not state explicitly. A few proofs have errors, by relying on assumptions that are intuitive but not explicitly proven. Mathematician and historian W. W. Rouse Ball put the criticisms in perspective, remarking that "the fact that for two thousand years [the Elements ]

5859-409: 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

5952-576: The internal angles of a triangle. Infinitely many triangles have 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

6045-588: 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

6138-415: 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

6231-402: The most difficult), leaving the others to the reader. Later editors such as Theon often interpolated their own proofs of these cases. Euclid's presentation was limited by the mathematical ideas and notations in common currency in his era, and this causes the treatment to seem awkward to the modern reader in some places. For example, there was no notion of an angle greater than two right angles,

6324-540: The most successful and influential textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first printing in 1482, the number reaching well over one thousand. For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements

6417-441: The number 1 was sometimes treated separately from other positive integers, and as multiplication was treated geometrically he did not use the product of more than 3 different numbers. The geometrical treatment of number theory may have been because the alternative would have been the extremely awkward Alexandrian system of numerals . The presentation of each result is given in a stylized form, which, although not invented by Euclid,

6510-407: 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

6603-411: 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

6696-485: The product of height and base length. In Euclidean geometry , any two points determine 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

6789-399: The propositions. The books cover plane and solid Euclidean geometry , elementary number theory , and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science , and its logical rigor was not surpassed until the 19th century. Euclid's Elements has been referred to as

6882-434: 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

6975-721: 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

7068-516: 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,

7161-412: 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

7254-591: 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

7347-498: The six books of Euclid at sight". Edna St. Vincent Millay wrote in her sonnet " Euclid alone has looked on Beauty bare ", "O blinding hour, O holy, terrible day, / When first the shaft into his vision shone / Of light anatomized!". Albert Einstein recalled a copy of the Elements and a magnetic compass as two gifts that had a great influence on him as a boy, referring to the Euclid as the "holy little geometry book". The success of

7440-609: The text. Also of importance are the scholia , or annotations to the text. These additions, which often distinguished themselves from the main text (depending on the manuscript), gradually accumulated over time as opinions varied upon what was worthy of explanation or further study. The Elements is still considered a masterpiece in the application of logic to mathematics . In historical context, it has proven enormously influential in many areas of science . Scientists Nicolaus Copernicus , Johannes Kepler , Galileo Galilei , Albert Einstein and Sir Isaac Newton were all influenced by

7533-488: 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),

7626-491: 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

7719-402: 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

7812-439: 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}} ,

7905-1279: 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

7998-419: 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

8091-419: The types of problems encountered in the first four books of the Elements . Some scholars have tried to find fault in Euclid's use of figures in his proofs, accusing him of writing proofs that depended on the specific figures drawn rather than the general underlying logic, especially concerning Proposition II of Book I. However, Euclid's original proof of this proposition, is general, valid, and does not depend on

8184-479: The use of letters to refer to figures. Other similar works are also reported to have been written by Theudius of Magnesia , Leon , and Hermotimus of Colophon. In the 4th century AD, Theon of Alexandria produced an edition of Euclid which was so widely used that it became the only surviving source until François Peyrard 's 1808 discovery at the Vatican of a manuscript not derived from Theon's. This manuscript,

8277-1702: 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 |}

8370-939: Was made of a Greek-to-Latin translation from the 12th century at Palermo, Sicily. The name of the translator is not known other than he was an anonymous medical student from Salerno who was visiting Palermo in order to translate the Almagest to Latin. The Euclid manuscript is extant and quite complete. After the translation by Adelard of Bath (known as Adelard I), there was a flurry of translations from Arabic. Notable translators in this period include Herman of Carinthia who wrote an edition around 1140, Robert of Chester (his manuscripts are referred to collectively as Adelard II, written on or before 1251), Johannes de Tinemue, possibly also known as John of Tynemouth (his manuscripts are referred to collectively as Adelard III), late 12th century, and Gerard of Cremona (sometime after 1120 but before 1187). The exact details concerning these translations

8463-413: Was not uncommon in ancient times to attribute to celebrated authors works that were not written by them. It is by these means that the apocryphal books XIV and XV of the Elements were sometimes included in the collection. The spurious Book XIV was probably written by Hypsicles on the basis of a treatise by Apollonius . The book continues Euclid's comparison of regular solids inscribed in spheres, with

8556-460: Was required of all students. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read. Scholars believe that the Elements is largely a compilation of propositions based on books by earlier Greek mathematicians. Proclus (412–485 AD), a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on

8649-538: Was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose." Later editors have added Euclid's implicit axiomatic assumptions in their list of formal axioms. For example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points. Known errors in Euclid date to at least 1882, when Pasch published his missing axiom . Early attempts to find all

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