Let p be a point inside a right triangle. Draw in the shortest paths to each side of the triangle and sum up the lengths. What is the average of such sums over all interior points?
I solved the problem using calculus, but I'm wondering if there's a faster, elementary solution
$$ \frac{\int_0^b \int_0^{ax/b} \left(a-x + y+\frac{ax-by}{\sqrt{a^2+b^2}} \right) \mathrm{d}y \, \mathrm{d}x}{ab/2} = \frac{a}{3} + \frac{b}{3} + \frac{ab}{3c}$$
Somewhat related: https://en.wikipedia.org/wiki/Viviani%27s_theorem