Pythagorean theorem (test post)

The Pythagorean theorem states that in a right triangle with legs of length \(a\) and \(b\) and hypotenuse of length \(c\), \[ a^2 + b^2 = c^2. \]

A simple example is \(a=3\), \(b=4\), \(c=5\), since \(3^2 + 4^2 = 9 + 16 = 25 = 5^2\).

For a right triangle, the incircle (the largest circle that fits inside it, tangent to all three sides) has radius \[ r = \frac{a+b-c}{2}. \] For \(a=3\), \(b=4\), \(c=5\), this gives \(r = \frac{3+4-5}{2} = 1\), so the incircle has area \(\pi r^2 = \pi\) and circumference \(2\pi r = 2\pi\).

Note that \(r\) is always positive, since \(a+b>c\) by the triangle inequality.