A little fun with the Erdős-Mordell Inequality Theorem

One of my favorite subjects in mathematics is Geometry.  You would think that, after 2,000 years or so, we would know all there is to know about the subject.  Of course, you would be horribly wrong.  Take, for example, this fun theorem that usually isn’t mentioned in a high school Geometry course.  Originally proposed by the legendary Paul Erdős in 1935, it was later proven by in 1937 by Louis Mordell and D. F. Barrow.

The Erdős-Mordell Inequality Theorem:  If P is a point inside of ΔABC, and P_A, P_{B and} P_C are the feet of the perpendiculars from P upon the respective sides BC, CA, and AB, then

Here’s the fun part, can you prove it?  It doesn’t hurt to start with a simpler case.  Consider an equilateral triangle where P is the circumcenter.  Consider an isosceles triangle where P is the circumcenter.  Can you prove it for these simpler cases?  Yes?  Good.  Now, can you prove it for the general case?  It isn’t as easy as it looks.

Note: diagram and Inequality provided at :Weisstein, Eric W. “Erdős-Mordell Theorem.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Erdos-MordellTheorem.html