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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.
Set square shaped as 45° - 45° - 90° triangle The side lengths of a 45° - 45° - 90° triangle 45° - 45° - 90° right triangle of hypotenuse length 1.. In plane geometry, dividing a square along its diagonal results in two isosceles right triangles, each with one right angle (90°, π / 2 radians) and two other congruent angles each measuring half of a right angle (45°, or ...
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle. [15] The incircle radius is no greater than one-ninth the sum of the altitudes. [16]: 289
Among isosceles triangles, the ratio of inradius to side length is maximized for the triangle formed by two reflected copies of the Kepler triangle, sharing the longer of their two legs. [53] The same isosceles triangle maximizes the ratio of the radius of a semicircle on its base to its perimeter. [54]
The reverse triangle inequality is an equivalent alternative formulation of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is: [19] Any side of a triangle is greater than or equal to the difference between the other two sides. In the case of a normed vector space, the statement is:
Given triangle sides b and c and angle γ there are sometimes two solutions for a. The theorem is used in solution of triangles , i.e., to find (see Figure 3): the third side of a triangle if two sides and the angle between them is known: c = a 2 + b 2 − 2 a b cos γ ; {\displaystyle c={\sqrt {a^{2}+b^{2}-2ab\cos \gamma }}\,;}
In any triangle, the distance along the boundary of the triangle from a vertex to the point on the opposite edge touched by an excircle equals the semiperimeter. The semiperimeter is used most often for triangles; the formula for the semiperimeter of a triangle with side lengths a, b, c = + +.
A triangle with sides a, b, and c. In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . Letting be the semiperimeter of the triangle, = (+ +), the area is [1]