This definitely works one way:
If the diagonals bisect each other then the point P exists. In fact it is very easy to show that P is the point of intersection found.
The problem with the statement is it isn't true the first way:
The point P can exist even if the diagonals don't bisect each other.
The simplest counterexample I can think of is a kite with AB=AD and BC=CD but AB≠BC. The diagonals don't intersect but P is the midpoint of AC we get four equal area triangles.
In general Let B and D be any points on the opposite sides of and equidistant from the line containing AC. P will be the midpoint of AC but need not be the midpoint of BD.
I was going to try to restate the problem but I can't fathom what the author of the problem was thinking.
Posted by Jer
on 2013-06-16 22:34:58