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from wikipedia

1. 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).
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)
The inradius of an isosceles triangle is given by

(5)
The mean of is given by

2. (6)


(7)
so the geometric centroid is


(8)


(9)
or 2/3 the way from its 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


(10)


(11)
so the area is


(12)
(13)


(14)


(15)

3. Erecting similar isosceles triangles on the edges of an initial triangle gives another
triangle such that , , and concur. The triangles are therefore perspective
triangles.
No set of points in the plane can determine only isosceles triangles.
SEE ALSO: 30-60-90 Triangle, Acute Triangle, Equilateral Triangle, Golden Gnomon, Golden
Triangle, Isosceles Right Triangle, Isosceles Tetrahedron, Isoscelizer, Kiepert Parabola, Obtuse
Triangle, Petr-Neumann-Douglas Theorem, Point Picking, Pons Asinorum, Right Triangle,
Scalene Triangle, Steiner-Lehmus Theorem