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Isosceles Triangle

An isosceles triangle is a triangle with (at least) two equal sides. In the figure above, the two equal sides have length and the remaining side has length . This property is equivalent to two angles of the triangle being equal. An isosceles triangle therefore has both two equal sides and two equal angles. The name derives from the Greek iso (same) and skelos (leg).

In diagrams, the two equal sides can be indicated by matching hatch marks, and the two equal angles by matching angle markers.

A triangle with all sides equal is called an equilateral triangle, and a triangle with no sides equal is called a scalene triangle. An equilateral triangle is therefore a special case of an isosceles triangle having not just two, but all three sides and angles equal. Another special case of an isosceles triangle is the isosceles right triangle.

The height of the isosceles triangle illustrated above can be found from the Pythagorean theorem as

(1)

The area is therefore given by

			(2)
			(3)
			(4)
			(5)
			(6)

where the inradius is given by

(7)

and the circumradius by

(8)

The mean of is given by

			(9)
			(10)

so the geometric centroid is

			(11)
			(12)

or 2/3 the way from the vertex (Gearhart and Schulz 1990).

Considering the angle at the apex of the triangle and writing instead of , there is a surprisingly simple relationship between the area and vertex angle . As shown in the above diagram, simple trigonometry gives

			(13)
			(14)

so the area is

			(15)
			(16)
			(17)
			(18)

Erecting similar isosceles triangles on the edges of an initial triangle gives another triangle such that , , and concur. The triangles are therefore perspective triangles.