Please consider the attached figure, given any acute where and are three altitudes from the vertices and . And is the triangle's orthocenter. Also let , and . Also note that and .
By definition, we have
Consider the right triangle , we have , which implies that
Now consider the two right triangles and , we have and , hence
From here, let's consider the right triangle , we have:
. But , we then have
which implies that
Use the quantities we obtained for and , we have