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69-604: The valley has the shape of an isosceles triangle , turned with its apex to the southeast. It has a length of 16 km and a width of up to 10 km. It spans a territory of 106 km with an altitude of 620–700 m. It is situated in the center of the Ihtimanska Sredna Gora, the western part of the Sredna Gora mountain range. The Ihtiman valley is enclosed between the Vakarel Mountain to

138-532: A {\displaystyle a} and t {\displaystyle t} exists. The Steiner–Lehmus theorem states that every triangle with two angle bisectors of equal lengths is isosceles. It was formulated in 1840 by C. L. Lehmus . Its other namesake, Jakob Steiner , was one of the first to provide a solution. Although originally formulated only for internal angle bisectors, it works for many (but not all) cases when, instead, two external angle bisectors are equal. The 30-30-120 isosceles triangle makes

207-399: A {\displaystyle a} is T = 3 4 a 2 . {\displaystyle T={\frac {\sqrt {3}}{4}}a^{2}.} The formula may be derived from the formula of an isosceles triangle by Pythagoras theorem : the altitude h {\displaystyle h} of a triangle is the square root of the difference of squares of a side and half of

276-504: A boundary case for this variation of the theorem, as it has four equal angle bisectors (two internal, two external). The inradius and circumradius formulas for an isosceles triangle may be derived from their formulas for arbitrary triangles. The radius of the inscribed circle of an isosceles triangle with side length a {\displaystyle a} , base b {\displaystyle b} , and height h {\displaystyle h} is: The center of

345-578: A right triangle , the median from the hypotenuse (that is, the line segment from the midpoint of the hypotenuse to the right-angled vertex) divides the right triangle into two isosceles triangles. This is because the midpoint of the hypotenuse is the center of the circumcircle of the right triangle, and each of the two triangles created by the partition has two equal radii as two of its sides. Similarly, an acute triangle can be partitioned into three isosceles triangles by segments from its circumcenter, but this method does not work for obtuse triangles, because

414-443: A base . Since the base and the legs are equal, the height is: h = a 2 − a 2 4 = 3 2 a . {\displaystyle h={\sqrt {a^{2}-{\frac {a^{2}}{4}}}}={\frac {\sqrt {3}}{2}}a.} In general, the area of a triangle is half the product of its base and height. The formula of the area of an equilateral triangle can be obtained by substituting

483-415: A circle with a certain radius, placing the point of the compass on the circle, and drawing another circle with the same radius; the two circles will intersect in two points. An equilateral triangle can be constructed by taking the two centers of the circles and the points of intersection. An alternative way to construct an equilateral triangle is by using Fermat prime . A Fermat prime is a prime number of

552-413: A corollary of this, the equilateral triangle has the smallest ratio of the circumradius R {\displaystyle R} to the inradius r {\displaystyle r} of any triangle. That is: R ≥ 2 r . {\displaystyle R\geq 2r.} Pompeiu's theorem states that, if P {\displaystyle P} is an arbitrary point in

621-514: A family of polyhedra incorporating a band of alternating triangles. When the antiprism is uniform , its bases are regular and all triangular faces are equilateral. As a generalization, the equilateral triangle belongs to the infinite family of n {\displaystyle n} - simplexes , with n = 2 {\displaystyle n=2} . Equilateral triangles have frequently appeared in man-made constructions and in popular culture. In architecture, an example can be seen in

690-466: A formula for the area of the polygon as a function of its side lengths, even for cyclic polygons that do not contain their circumcenters. This formula generalizes Heron's formula for triangles and Brahmagupta's formula for cyclic quadrilaterals . Either diagonal of a rhombus divides it into two congruent isosceles triangles. Similarly, one of the two diagonals of a kite divides it into two isosceles triangles, which are not congruent except when

759-474: A given perimeter is equilateral. That is, for perimeter p {\displaystyle p} and area T {\displaystyle T} , the equality holds for the equilateral triangle: p 2 = 12 3 T . {\displaystyle p^{2}=12{\sqrt {3}}T.} The radius of the circumscribed circle is: R = a 3 , {\displaystyle R={\frac {a}{\sqrt {3}}},} and

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828-526: A given triangle, that triangle must be isosceles. The area T {\displaystyle T} of an isosceles triangle can be derived from the formula for its height, and from the general formula for the area of a triangle as half the product of base and height: The same area formula can also be derived from Heron's formula for the area of a triangle from its three sides. However, applying Heron's formula directly can be numerically unstable for isosceles triangles with very sharp angles, because of

897-526: A plane, known as the trigonal planar molecular geometry . In the Thomson problem , concerning the minimum-energy configuration of n {\displaystyle n} charged particles on a sphere, and for the Tammes problem of constructing a spherical code maximizing the smallest distance among the points, the best solution known for n = 3 {\displaystyle n=3} places

966-485: A point P {\displaystyle P} in the interior of an equilateral triangle, the ratio of the sum of its distances from the vertices to the sum of its distances from the sides is greater than or equal to 2, equality holding when P {\displaystyle P} is the centroid. In no other triangle is there a point for which this ratio is as small as 2. This is the Erdős–Mordell inequality ;

1035-435: A straight line and place the point of the compass on one end of the line, then swing an arc from that point to the other point of the line segment; repeat with the other side of the line, which connects the point where the two arcs intersect with each end of the line segment in the aftermath. If three equilateral triangles are constructed on the sides of an arbitrary triangle, either all outward or inward, by Napoleon's theorem

1104-467: A stronger variant of it is Barrow's inequality , which replaces the perpendicular distances to the sides with the distances from P {\displaystyle P} to the points where the angle bisectors of ∠ A P B {\displaystyle \angle APB} , ∠ B P C {\displaystyle \angle BPC} , and ∠ C P A {\displaystyle \angle CPA} cross

1173-500: A stylized image of a mountain island. They also have been used in designs with religious or mystic significance, for instance in the Sri Yantra of Hindu meditational practice . If a cubic equation with real coefficients has three roots that are not all real numbers , then when these roots are plotted in the complex plane as an Argand diagram they form vertices of an isosceles triangle whose axis of symmetry coincides with

1242-615: Is 548 mm. It is drained by the river Mativir , a right tributary of the Topolnitsa of the Maritsa river basin. The soils are mainly chernozem and alluvial. Administratively, it falls in Sofia Province on the territory of Ihtiman Municipality . Apart from the town of Ihtiman, there are eight villages — Boeritsa , Venkovets , Verinsko , Zhivkovo , Mirovo , Polyantsi , Stambolovo and Chernyovo . The valley

1311-420: Is a triangle that has three equal sides. It is a special case of an isosceles triangle in the modern definition, stating that an isosceles triangle is defined at least as having two equal sides. Based on the modern definition, this leads to an equilateral triangle in which one of the three sides may be considered its base. The follow-up definition above may result in more precise properties. For example, since

1380-536: Is also the center of the circumcircle that passes through the three vertices). In an isosceles triangle with exactly two equal sides, these three points are distinct, and (by symmetry) all lie on the symmetry axis of the triangle, from which it follows that the Euler line coincides with the axis of symmetry. The incenter of the triangle also lies on the Euler line, something that is not true for other triangles. If any two of an angle bisector, median, or altitude coincide in

1449-580: Is an isosceles triangle that is acute, but less so than the equilateral triangle; its height is proportional to 5/8 of its base. The Egyptian isosceles triangle was brought back into use in modern architecture by Dutch architect Hendrik Petrus Berlage . Warren truss structures, such as bridges, are commonly arranged in isosceles triangles, although sometimes vertical beams are also included for additional strength. Surfaces tessellated by obtuse isosceles triangles can be used to form deployable structures that have two stable states: an unfolded state in which

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1518-520: Is called a deltahedron . There are eight strictly convex deltahedra: three of the five Platonic solids ( regular tetrahedron , regular octahedron , and regular icosahedron ) and five of the 92 Johnson solids ( triangular bipyramid , pentagonal bipyramid , snub disphenoid , triaugmented triangular prism , and gyroelongated square bipyramid ). More generally, all Johnson solids have equilateral triangles among their faces, though most also have other other regular polygons . The antiprisms are

1587-410: Is equilateral. If the two equal sides have length a {\displaystyle a} and the other side has length b {\displaystyle b} , then the internal angle bisector t {\displaystyle t} from one of the two equal-angled vertices satisfies as well as and conversely, if the latter condition holds, an isosceles triangle parametrized by

1656-684: Is located a 12.2 km stretch of the third class III-822 road Samokov – Novo Selo –Ihtiman. The center of the valley is traversed by railway line No. 1 Kalotina–Sofia–Plovdiv– Svilengrad served by the Bulgarian State Railways . The climatic and soil conditions of the Ihtiman Valley are favourable for agriculture, mainly cereals and potatoes, as well as animal husbandry. In Ihtiman there is metallurgic, machine building and textile industry. The oldest golf course in Bulgaria

1725-485: Is not isosceles (having three unequal sides) is called scalene . "Isosceles" is made from the Greek roots "isos" (equal) and "skelos" (leg). The same word is used, for instance, for isosceles trapezoids , trapezoids with two equal sides, and for isosceles sets , sets of points every three of which form an isosceles triangle. In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and

1794-488: Is respectively obtuse, right or acute. In Edwin Abbott 's book Flatland , this classification of shapes was used as a satire of social hierarchy : isosceles triangles represented the working class , with acute isosceles triangles higher in the hierarchy than right or obtuse isosceles triangles. As well as the isosceles right triangle , several other specific shapes of isosceles triangles have been studied. These include

1863-474: Is rooted in Euclid's lack of recognition of the concept of betweenness and the resulting ambiguity of inside versus outside of figures. Equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon , occasionally known as the regular triangle . It

1932-399: Is served by several roads of the national network, as well as local roads. Its eastern reaches are traversed by a 11.9 km stretch Trakia motorway in direction northwest–southeast. Parallel to the motorway runs a 16.3 km section of the first class I-8 road Kalotina – Sofia – Plovdiv – Kapitan Andreevo , both of them part of the strategic European route E80 . In direction west–east

2001-448: Is situated in the valley. Isosceles triangle In geometry , an isosceles triangle ( / aɪ ˈ s ɒ s ə l iː z / ) is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case . Examples of isosceles triangles include

2070-423: Is the special case of an isosceles triangle by modern definition, creating more special properties. The equilateral triangle can be found in various tilings , and in polyhedrons such as the deltahedron and antiprism . It appears in real life in popular culture, architecture, and the study of stereochemistry resembling the molecular known as the trigonal planar molecular geometry . An equilateral triangle

2139-707: The Calabi triangle (a triangle with three congruent inscribed squares), the golden triangle and golden gnomon (two isosceles triangles whose sides and base are in the golden ratio ), the 80-80-20 triangle appearing in the Langley's Adventitious Angles puzzle, and the 30-30-120 triangle of the triakis triangular tiling . Five Catalan solids , the triakis tetrahedron , triakis octahedron , tetrakis hexahedron , pentakis dodecahedron , and triakis icosahedron , each have isosceles-triangle faces, as do infinitely many pyramids and bipyramids . For any isosceles triangle,

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2208-486: The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus . The theorem that the base angles of an isosceles triangle are equal appears as Proposition I.5 in Euclid. This result has been called the pons asinorum (the bridge of asses) or the isosceles triangle theorem. Rival explanations for this name include the theory that it is because the diagram used by Euclid in his demonstration of

2277-482: The Pythagorean theorem using the fact that the altitude bisects the base and partitions the isosceles triangle into two congruent right triangles. The Euler line of any triangle goes through the triangle's orthocenter (the intersection of its three altitudes), its centroid (the intersection of its three medians), and its circumcenter (the intersection of the perpendicular bisectors of its three sides, which

2346-470: The Sierpiński triangle (a fractal shape constructed from an equilateral triangle by subdividing recursively into smaller equilateral triangles) and Reuleaux triangle (a curved triangle with constant width , constructed from an equilateral triangle by rounding each of its sides). Equilateral triangles may also form a polyhedron in three dimensions. A polyhedron whose faces are all equilateral triangles

2415-421: The isoperimetric inequality This is a strict inequality for isosceles triangles with sides unequal to the base, and becomes an equality for the equilateral triangle. The area, perimeter, and base can also be related to each other by the equation If the base and perimeter are fixed, then this formula determines the area of the resulting isosceles triangle, which is the maximum possible among all triangles with

2484-507: The isosceles right triangle , the golden triangle , and the faces of bipyramids and certain Catalan solids . The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics . Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings. The two equal sides are called

2553-478: The perimeter of an isosceles triangle is the sum of its two legs and base, the equilateral triangle is formulated as three times its side. The internal angle of an equilateral triangle are equal, 60°. Because of these properties, the equilateral triangles are regular polygons . The cevians of an equilateral triangle are all equal in length, resulting in the median and angle bisector being equal in length, considering those lines as their altitude depending on

2622-419: The triangle inequality that the sum of any two of them is greater than the third. If P {\displaystyle P} is on the circumcircle then the sum of the two smaller ones equals the longest and the triangle has degenerated into a line, this case is known as Van Schooten's theorem . A packing problem asks the objective of n {\displaystyle n} circles packing into

2691-438: The altitude formula. Another way to prove the area of an equilateral triangle is by using the trigonometric function . The area of a triangle is formulated as the half product of base and height and the sine of an angle. Because all of the angles of an equilateral triangle are 60°, the formula is as desired. A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with

2760-408: The angle at its apex. In Euclidean geometry , the base angles can not be obtuse (greater than 90°) or right (equal to 90°) because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. Since a triangle is obtuse or right if and only if one of its angles is obtuse or right, respectively, an isosceles triangle is obtuse, right or acute if and only if its apex angle

2829-401: The apex. For any isosceles triangle, there is a unique square with one side collinear with the base of the triangle and the opposite two corners on its sides. The Calabi triangle is a special isosceles triangle with the property that the other two inscribed squares, with sides collinear with the sides of the triangle, are of the same size as the base square. A much older theorem, preserved in

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2898-396: The base's choice. When the equilateral triangle is flipped across its altitude or rotated around its center for one-third of a full turn, its appearance is unchanged; it has the symmetry of a dihedral group D 3 {\displaystyle \mathrm {D} _{3}} of order six. Other properties are discussed below. The area of an equilateral triangle with edge length

2967-583: The centers of those equilateral triangles themselves form an equilateral triangle. Notably, the equilateral triangle tiles the Euclidean plane with six triangles meeting at a vertex; the dual of this tessellation is the hexagonal tiling . Truncated hexagonal tiling , rhombitrihexagonal tiling , trihexagonal tiling , snub square tiling , and snub hexagonal tiling are all semi-regular tessellations constructed with equilateral triangles. Other two-dimensional objects built from equilateral triangles include

3036-430: The circle lies on the symmetry axis of the triangle, this distance above the base. An isosceles triangle has the largest possible inscribed circle among the triangles with the same base and apex angle, as well as also having the largest area and perimeter among the same class of triangles. The radius of the circumscribed circle is: The center of the circle lies on the symmetry axis of the triangle, this distance below

3105-405: The circumcenter lies outside the triangle. Generalizing the partition of an acute triangle, any cyclic polygon that contains the center of its circumscribed circle can be partitioned into isosceles triangles by the radii of this circle through its vertices. The fact that all radii of a circle have equal length implies that all of these triangles are isosceles. This partition can be used to derive

3174-587: The cross-section of the Gateway Arch and the surface of the Vegreville egg . It appears in the flag of Nicaragua and the flag of the Philippines . It is a shape of a variety of road signs , including the yield sign . The equilateral triangle occurs in the study of stereochemistry . It can be described as the molecular geometry in which one atom in the center connects three other atoms in

3243-440: The feet of the altitudes ), and the only triangle whose Steiner inellipse is a circle (specifically, the incircle). The triangle of the largest area of all those inscribed in a given circle is equilateral, and the triangle of the smallest area of all those circumscribed around a given circle is also equilateral. It is the only regular polygon aside from the square that can be inscribed inside any other regular polygon. Given

3312-420: The following six line segments coincide: Their common length is the height h {\displaystyle h} of the triangle. If the triangle has equal sides of length a {\displaystyle a} and base of length b {\displaystyle b} , the general triangle formulas for the lengths of these segments all simplify to This formula can also be derived from

3381-438: The form 2 2 k + 1 , {\displaystyle 2^{2^{k}}+1,} wherein k {\displaystyle k} denotes the non-negative integer , and there are five known Fermat primes: 3, 5, 17, 257, 65537. A regular polygon is constructible by compass and straightedge if and only if the odd prime factors of its number of sides are distinct Fermat primes. To do so geometrically, draw

3450-650: The foundation is composed of Neogene – Quaternary sediments. The valley was formed as a result of tectonic subsidence at the end of the Upper Pliocene and the Quaternary. The climate is temperate continental with frequent temperature inversions and mists in winter. The average annual temperature is 9.1 °C, with an average of -2.3 °C in January and 19.5 °C in July. The mean annual precipitation

3519-410: The horizontal (real) axis. This is because the complex roots are complex conjugates and hence are symmetric about the real axis. In celestial mechanics , the three-body problem has been studied in the special case that the three bodies form an isosceles triangle, because assuming that the bodies are arranged in this way reduces the number of degrees of freedom of the system without reducing it to

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3588-596: The kite is a rhombus. Isosceles triangles commonly appear in architecture as the shapes of gables and pediments . In ancient Greek architecture and its later imitations, the obtuse isosceles triangle was used; in Gothic architecture this was replaced by the acute isosceles triangle. In the architecture of the Middle Ages , another isosceles triangle shape became popular: the Egyptian isosceles triangle. This

3657-407: The legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs and base. Every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base. The two angles opposite the legs are equal and are always acute , so the classification of

3726-407: The near-cancellation between the semiperimeter and side length in those triangles. If the apex angle ( θ ) {\displaystyle (\theta )} and leg lengths ( a ) {\displaystyle (a)} of an isosceles triangle are known, then the area of that triangle is: This is a special case of the general formula for the area of a triangle as half

3795-564: The northwest and the ridges of Belitsa to the northeast, Vetren to the southeast, and Septemvriyski to the south–southwest. The latter separates it from the Kostenets–Dolna Banya Valley to the south. The slopes of the Ihtiman Valley have uneven gradient and size, ranging from longer and oblique at the Vakarel Mountain, to steep Septemvriyski ridge. Its center is an extensive flat accumulative field. The slopes are built up of gneiss , granite , sandstone and conglomerates , while

3864-518: The plane of an equilateral triangle A B C {\displaystyle ABC} but not on its circumcircle , then there exists a triangle with sides of lengths P A {\displaystyle PA} , P B {\displaystyle PB} , and P C {\displaystyle PC} . That is, P A {\displaystyle PA} , P B {\displaystyle PB} , and P C {\displaystyle PC} satisfy

3933-429: The product of two sides times the sine of the included angle. The perimeter p {\displaystyle p} of an isosceles triangle with equal sides a {\displaystyle a} and base b {\displaystyle b} is just As in any triangle, the area T {\displaystyle T} and perimeter p {\displaystyle p} are related by

4002-442: The radius of the inscribed circle is half of the circumradius: r = 3 6 a . {\displaystyle r={\frac {\sqrt {3}}{6}}a.} The theorem of Euler states that the distance t {\displaystyle t} between circumradius and inradius is formulated as t 2 = R ( R − 2 r ) {\displaystyle t^{2}=R(R-2r)} . As

4071-479: The result resembles a bridge, or because this is the first difficult result in Euclid, and acts to separate those who can understand Euclid's geometry from those who cannot. A well-known fallacy is the false proof of the statement that all triangles are isosceles , first published by W. W. Rouse Ball in 1892, and later republished in Lewis Carroll 's posthumous Lewis Carroll Picture Book . The fallacy

4140-427: The same base and perimeter. On the other hand, if the area and perimeter are fixed, this formula can be used to recover the base length, but not uniquely: there are in general two distinct isosceles triangles with given area T {\displaystyle T} and perimeter p {\displaystyle p} . When the isoperimetric inequality becomes an equality, there is only one such triangle, which

4209-506: The sides ( A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} being the vertices). There are numerous other triangle inequalities that hold equality if and only if the triangle is equilateral. The equilateral triangle can be constructed in different ways by using circles. The first proposition in the Elements first book by Euclid . Start by drawing

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4278-432: The sides and altitude h {\displaystyle h} , d + e + f = h , {\displaystyle d+e+f=h,} independent of the location of P {\displaystyle P} . An equilateral triangle may have integer sides with three rational angles as measured in degrees, known for the only acute triangle that is similar to its orthic triangle (with vertices at

4347-744: The smallest possible equilateral triangle . The optimal solutions show n < 13 {\displaystyle n<13} that can be packed into the equilateral triangle, but the open conjectures expand to n < 28 {\displaystyle n<28} . Morley's trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. Viviani's theorem states that, for any interior point P {\displaystyle P} in an equilateral triangle with distances d {\displaystyle d} , e {\displaystyle e} , and f {\displaystyle f} from

4416-445: The solved Lagrangian point case when the bodies form an equilateral triangle. The first instances of the three-body problem shown to have unbounded oscillations were in the isosceles three-body problem. Long before isosceles triangles were studied by the ancient Greek mathematicians , the practitioners of Ancient Egyptian mathematics and Babylonian mathematics knew how to calculate their area. Problems of this type are included in

4485-515: The surface expands to a cylindrical column, and a folded state in which it folds into a more compact prism shape that can be more easily transported. The same tessellation pattern forms the basis of Yoshimura buckling , a pattern formed when cylindrical surfaces are axially compressed, and of the Schwarz lantern , an example used in mathematics to show that the area of a smooth surface cannot always be accurately approximated by polyhedra converging to

4554-481: The surface. In graphic design and the decorative arts , isosceles triangles have been a frequent design element in cultures around the world from at least the Early Neolithic to modern times. They are a common design element in flags and heraldry , appearing prominently with a vertical base, for instance, in the flag of Guyana , or with a horizontal base in the flag of Saint Lucia , where they form

4623-406: The third side is called the base . The angle included by the legs is called the vertex angle and the angles that have the base as one of their sides are called the base angles . The vertex opposite the base is called the apex . In the equilateral triangle case, since all sides are equal, any side can be called the base. Whether an isosceles triangle is acute, right or obtuse depends only on

4692-451: The triangle as acute, right, or obtuse depends only on the angle between its two legs. Euclid defined an isosceles triangle as a triangle with exactly two equal sides, but modern treatments prefer to define isosceles triangles as having at least two equal sides. The difference between these two definitions is that the modern version makes equilateral triangles (with three equal sides) a special case of isosceles triangles. A triangle that

4761-458: The works of Hero of Alexandria , states that, for an isosceles triangle with base b {\displaystyle b} and height h {\displaystyle h} , the side length of the inscribed square on the base of the triangle is For any integer n ≥ 4 {\displaystyle n\geq 4} , any triangle can be partitioned into n {\displaystyle n} isosceles triangles. In

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