Consider an acute triangle . Let denote the orthocenter and let , , be the feet of the perpendiculars of , , and respectfully.
Let , , , represent the area of , , and , respectfully. So we have that
Now we also have that
(By substituting previous values of , , , )
Now we note that:
By substituting each new altitude representation into our previous result gives us
Which is what we wanted!
Notice if is an obtuse triangle our relation no longer holds since the orthocenter lies outside the triangle .
Nevertheless, we can now consider the triangle , which has orthocenter , thereby reducing to the previously proven case!