# Fun Identity in Right Triangle

### Source

### Solution

With multiple applications of he Pythagorean Theorem,

$\begin{align} p^2+q^2&=h^2=xy\\ x^2&=m^2+p^2\\ y^2&=n^2+q^2\\ 2xy&=2h^2\\ a^2&=(x+y)^2=x^2+y^2+2xy\\ &=m^2+p^2+n^2+q^2+2h^2\\ &=m^2+n^2+(p^2+q^2)+2h^2\\ &=m^2+n^2+3h^2. \end{align}$

### Acknowledgment

I borrowed this problem from the Peru Geometrico facebook group. The problem is by Miguel Ocha Sanchez. There is a good half dozen additional solutions.

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