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

$h^2= \sqrt{a^2-(\frac{b}{2})^2}$

The area is:

$A= \sqrt{\frac{a^2b^2}{4}-\frac{b^4}{16}}$

The perimeter is:

$P= 2a+b$

Trigonometric functions of half angles in a triangle

$Sin= \sqrt{\frac{1}{2}-\frac{b}{4a}}$

$Cos= \sqrt{\frac{b}{4a}+\frac{1}{2}}$

$Tan= \sqrt{\frac{2a-b}{2a+b}}$

$Csc= \sqrt{\frac{4a}{2a-b}}$

$Sec= \sqrt{\frac{4a}{2a+b}}$

$Cot= \sqrt{\frac{2a+b}{2a-b}}$

Trigonometric functions of whole angles in a triangle

$Sin= \frac{opposite}{hypotenuse}=\frac{2A}{ba}= \sqrt{1-\frac{b^2}{4a^2}}$

$Cos= \frac{adjacent}{hypotenuse}= \frac{b}{2a}$

$Tan= \frac{opposite}{adjacent}= \frac{4A}{b^2} =\sqrt{\frac{4a^2}{b^2}-1}$

$Csc= \frac{hypotenuse}{opposite}= \frac{ba}{2A}= \sqrt{\frac{4a^2}{4a^2-b^2}}$

$Sec= \frac{hypotenuse}{adjacent}= \frac{2a}{b}$

$Cot= \frac{adjacent}{opposite}= \frac{b^2}{4A}= \sqrt{\frac{b^2}{4a^2- b^2}}$

CircumCircle

$R= \frac{a^2b}{\sqrt{4a^2b^2-b^4}}$

InCircle

$r= \frac{\sqrt{a^2b^2-\frac{b^4}{4}}}{2a+b}$