"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 is not isosceles (having three unequal sides) is called scalene. ""Isosceles"" is a compound word, made from the Greek roots ""isos"" (equal) and ""skelos"" (leg). The same word is used, for instance, for isosceles trapezoids, trapezoids with"
"this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Triangles can be classified according to the lengths of their sides: Hatch marks, also called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of ""ticks"", short line segments in the form of tally marks; two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is usually no more than 3"
"A triangle (""X"", ""Y"", ""Z"", ""u"", ""v"", ""w"") consists of 3 objects ""X"", ""Y"", and ""Z"", together with morphisms ""u"" : ""X"" → ""Y"", ""v"" : ""Y"" → ""Z"" and ""w"" : ""Z"" → ""X""[1]. Triangles are generally written in the unravelled form: Triangulated category In mathematics, a triangulated category is a category together with the additional structure of a ""translation functor"" and a class of ""distinguished triangles"". Prominent examples are the derived category of an abelian category and the stable homotopy category of spectra (more generally, the homotopy category of a stable ∞-category), both of which carry the structure"
"of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. That sum can equal the length of the third side only in the case of a degenerate triangle, one with collinear vertices. It is not possible for that sum to be less than the length of the third side. A triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality. Three given angles form a non-degenerate triangle (and indeed an infinitude of them) if and only if both"
"its sides be in a geometric progression. The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in an arithmetic progression. Let be the side length of a regular decagon inscribed in the unit circle, where ""φ"" is the golden ratio. Let be the side length of a regular hexagon in the unit circle, and let {2}}</math>}} be the side length of a regular pentagon in the unit circle. Then , so these three lengths form the sides of a right triangle. The same triangle forms half of a golden rectangle. It may also be found"
"""c"" is the length of the hypotenuse of a right triangle having sides ""a"" and ""b"". For instance, in the illustration to the left, the two squares in the Pythagorean tiling have side lengths 5 and 12 units long, and the side length of the tiles in the overlaying square tiling is 13, based on the Pythagorean triple (5,12,13). By overlaying a square grid of side length ""c"" onto the Pythagorean tiling, it may be used to generate a five-piece dissection of two unequal squares of sides ""a"" and ""b"" into a single square of side ""c"", showing that the"
"Isosceles triangle In geometry, an isosceles triangle 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 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"
"type of triangles: Triangular tiling In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}. Conway calls it a deltille, named from the triangular shape of the Greek letter delta (Δ). The triangular tiling can also be called a kishextille by a kis operation that adds a center point and triangles to replace the faces of a hextille. It is one of"
"sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle. The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely. The triangle sides ""a"", ""b"", ""c"" can"
"with angles α, β and γ is similar to a triangle with side lengths equal to sin α, sin β and sin γ. This triangle can be constructed by first constructing a circle of diameter 1, and inscribing in it two of the angles of the triangle. The length of the sides of that triangle will be sin α, sin β and sin γ. The side whose length is sin α is opposite to the angle whose measure is α, etc. The law of cosines, or cosine rule, connects the length of an unknown side of a triangle to the"
"of recognition of the concept of ""betweenness"" and the resulting ambiguity of ""inside"" versus ""outside"" of figures. Isosceles triangle In geometry, an isosceles triangle 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 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"
"of two sides of a triangle equals the altitude to the third side times the diameter ""D"" of the circumcircle: Suppose two adjacent but non-overlapping triangles share the same side of length ""f"" and share the same circumcircle, so that the side of length ""f"" is a chord of the circumcircle and the triangles have side lengths (""a"", ""b"", ""f"") and (""c"", ""d"", ""f""), with the two triangles together forming a cyclic quadrilateral with side lengths in sequence (""a"", ""b"", ""c"", ""d""). Then Let ""G"" be the centroid of a triangle with vertices ""A"", ""B"", and ""C"", and let ""P"""
"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 the works of Hero of Alexandria, states that, for an isosceles triangle with base formula_3 and height formula_1, the side length of the inscribed square on the base of the triangle is For any integer formula_35, any triangle can be partitioned into formula_36 isosceles triangles. In a right triangle, the median from the hypotenuse (that is, the line segment from the midpoint of"
"side of a triangle is the segment, falling entirely on and inside the triangle, of the line that perpendicularly bisects that side. The three perpendicular bisectors of a triangle's three sides intersect at the circumcenter (the center of the circle through the three vertices). Thus any line through a triangle's circumcenter and perpendicular to a side bisects that side. In an acute triangle the circumcenter divides the interior perpendicular bisectors of the two shortest sides in equal proportions. In an obtuse triangle the two shortest sides' perpendicular bisectors (extended beyond their opposite triangle sides to the circumcenter) are divided by"
"reflects a median in the angle bisector that passes through the same vertex, one obtains a symmedian. The three symmedians intersect in a single point, the symmedian point of the triangle. There are various standard methods for calculating the length of a side or the measure of an angle. Certain methods are suited to calculating values in a right-angled triangle; more complex methods may be required in other situations. In right triangles, the trigonometric ratios of sine, cosine and tangent can be used to find unknown angles and the lengths of unknown sides. The sides of the triangle are known"
"as shown in the upper part of the diagram, with ""BC"" the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length ""y"", the side ""AC"" of length ""x"" and the side ""AB"" of length ""a"", as seen in the lower diagram part. If ""x"" is increased by a small amount ""dx"" by extending the side ""AC"" slightly to ""D"", then ""y"" also increases by ""dy"". These form two sides of a triangle, ""CDE"", which (with ""E"" chosen so ""CE"" is perpendicular to the hypotenuse) is a right triangle approximately similar to ""ABC""."
"angle A"" and ""opposed to angle B"". If the lengths of all three sides of a right triangle are integers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple. As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the base then the other is height, so the area of a right triangle is one half the product of the two legs. As a formula the area ""T"" is where ""a"""
"""x"", then If two obtuse triangles have sides (""a, b, c"") and (""p, q, r"") with ""c"" and ""r"" being the respective longest sides, then The Calabi triangle, which is the only non-equilateral triangle for which the largest square that fits in the interior can be positioned in any of three different ways, is obtuse and isosceles with base angles 39.1320261...° and third angle 101.7359477...°. The equilateral triangle, with three 60° angles, is acute. The Morley triangle, formed from any triangle by the intersections of its adjacent angle trisectors, is equilateral and hence acute. The golden triangle is the isosceles"
"Right triangle A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the ""hypotenuse"" (side ""c"" in the figure). The sides adjacent to the right angle are called ""legs"" (or ""catheti"", singular: ""cathetus""). Side ""a"" may be identified as the side ""adjacent to angle B"" and ""opposed to"" (or ""opposite"") ""angle A"", while side ""b"" is the side ""adjacent to"
"Right triangle A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the ""hypotenuse"" (side ""c"" in the figure). The sides adjacent to the right angle are called ""legs"" (or ""catheti"", singular: ""cathetus""). Side ""a"" may be identified as the side ""adjacent to angle B"" and ""opposed to"" (or ""opposite"") ""angle A"", while side ""b"" is the side ""adjacent to"
"is 90°, leaving the remaining angle to be 30°. Right triangles whose sides are of integer lengths, with the sides collectively known as Pythagorean triples, possess angles that cannot all be rational numbers of degrees. (This follows from Niven's theorem.) They are most useful in that they may be easily remembered and any multiple of the sides produces the same relationship. Using Euclid's formula for generating Pythagorean triples, the sides must be in the ratio where ""m"" and ""n"" are any positive integers such that . There are several Pythagorean triples which are well-known, including those with sides in the"
"side length 3""a"" by the removal of six right square pyramids, one from each point. These pyramids have both base side length (""a"") and lateral side length (""e"") of ""a"", to form equilateral triangles. The base area is then ""a"". Note that this shape is exactly similar to half an octahedron or Johnson solid J. From the properties of square pyramids, we can now find the slant height, ""s"", and the height, ""h"", of the pyramid: The volume, ""V"", of the pyramid is given by: Because six pyramids are removed by truncation, there is a total lost volume of ""a""."
"Heronian triangle In geometry, a Heronian triangle is a triangle that has side lengths and area that are all integers. Heronian triangles are named after Hero of Alexandria. The term is sometimes applied more widely to triangles whose sides and area are all rational numbers, since one can rescale the sides by a common multiple to obtain a triangle that is Heronian in the above sense. Any right-angled triangle whose sidelengths are a Pythagorean triple is a Heronian triangle, as the side lengths of such a triangle are integers, and its area is also an integer, being half of the"
"a triangle, the angles can be derived if one knows the length of all sides, (see congruence), but the length of the sides cannot be derived based on all of the angles, not without knowing the length of at least one of the sides (a baseline) (see similarity). In 3D, when four or more angles are in play, locations can be calculated from ""n"" + 1 = 4 measured angles plus one known baseline or from just ""n"" + 1 = 4 measured sides. Multilateration is often more accurate for locating an object than approaches such as trilateration or triangulation,"
"Elongated triangular tiling In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated by rows of squares, and given Schläfli symbol {3,6}:e. Conway calls it a isosnub quadrille. There are 3 regular and 8 semiregular tilings in the plane. This tiling is similar to the snub square tiling which also has 3 triangles and two squares on a vertex, but in a different order. It is also the only convex uniform tiling that can't be created as a"
"appear frequently in architecture and design, for instance in the pediments and gables of buildings. The two equal sides are called 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 the triangle as acute, right, or obtuse depends"
"within a regular icosahedron of side length ""c"": the shortest line segment from any vertex ""V"" to the plane of its five neighbors has length ""a"", and the endpoints of this line segment together with any of the neighbors of ""V"" form the vertices of a right triangle with sides ""a"", ""b"", and ""c"". Special right triangle A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called"
"Kepler triangle A Kepler triangle is a right triangle with edge lengths in a geometric progression in which the common ratio is , where is the golden ratio, and can be written: formula_1, or approximately 1 : 1.272 : 1.618. The squares of the edges of this triangle are in geometric progression according to the golden ratio. Triangles with such ratios are named after the German mathematician and astronomer Johannes Kepler (1571–1630), who first demonstrated that this triangle is characterised by a ratio between its short side and hypotenuse equal to the golden ratio. Kepler triangles combine two key mathematical"
"satisfying the formula ; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides and is a right triangle, but is not a Pythagorean triple because is not an integer. Moreover, and do not have an integer common multiple because is irrational. Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system. It was discovered by Edgar James Banks shortly"
"according to This implies that the longest side in an obtuse triangle is the one opposite the obtuse-angled vertex. An acute triangle has three inscribed squares, each with one side coinciding with part of a side of the triangle and with the square's other two vertices on the remaining two sides of the triangle. (In a right triangle two of these are merged into the same square, so there are only two distinct inscribed squares.) However, an obtuse triangle has only one inscribed square, one of whose sides coincides with part of the longest side of the triangle. All triangles"
"the circumcenter is located on a side of the triangle, then the opposite angle is a right one. If the circumcenter is located inside the triangle, then 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 (i.e. forming a right angle with) 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"
"product of the two shorter sides of the triangle, at least one of which must be even. An example of a Heronian triangle which is not right-angled is the isosceles triangle with sidelengths 5, 5, and 6, whose area is 12. This triangle is obtained by joining two copies of the right-angled triangle with sides 3, 4, and 5 along the sides of length 4. This approach works in general, as illustrated in the adjacent picture. One takes a Pythagorean triple (""a"", ""b"", ""c""), with ""c"" being largest, then another one (""a"", ""d"", ""e""), with ""e"" being largest, constructs the"
"Tri-oval A tri-oval is a shape which derives its name from the two other shapes it most resembles, a triangle and an oval. Rather than meeting at sharp, definable angles as the sides of a triangle do, in a tri-oval these angles are instead rounded into smooth curves. While an oval has four turns, a tri-oval has six. More formally, according to the four-vertex theorem, every smooth simple closed curve has at least four vertices, points where its curvature reaches a local minimum or maximum. In a tri-oval, there are six such points, alternating between three minima and three maxima."