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In geometry, an isosceles triangle (/ aɪ ˈ s ɒ s ə l iː z /) is a triangle that has two sides of equal length or two angles of equal measure. 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.
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 is the special case of an isosceles triangle by modern definition, creating more special properties.
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, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
Position of some special triangles in an Euler diagram of types of triangles, using the definition that isosceles triangles have at least two equal sides, i.e. equilateral triangles are isosceles. A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas ...
Isosceles triangle with equal sides AB = AC divided into two right triangles by an altitude drawn from one of the two base angles. In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.
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 triangle in which the ratio of the duplicated side to the base side equals the golden ratio. It is acute, with ...
Duplicate the right triangle to form the isosceles triangle ACP. Construct the circle with center A and radius b, and its tangent h = BH through B. The tangent h forms a right angle with the radius b (Euclid's Elements: Book 3, Proposition 18; or see here), so the yellow triangle in Figure 8 is right. Apply the Pythagorean theorem to obtain
In the latter double inequality, the first part holds with equality if and only if the triangle is isosceles with an apex angle of at least 60°, and the last part holds with equality if and only if the triangle is isosceles with an apex angle of at most 60°. Thus both are equalities if and only if the triangle is equilateral. [7]: Thm. 1