Posted - 11/18/2013 : 18:31:21 Your proof is interesting. It shows that AE+BD+CF > 0.5(AB+BC+AC). However, as you find, this doesn't prove AE+CF+BD < AB+BC+CD. Here is a hint as to prove what you are asking: See that the triangles formed are right triangles and use the Pythagorean theorem to show that a leg (which are the altitudes) is always less than the hypotenuse. Once you show that, you can say that the sum of all three of those "legs" is less than the sum of all three of the "hypotenuses". If you need further help, please let us know.