I found this problem from the 1981 Canadian Math Society’s magazine, Crux Mathematicorum, to be quite challenging.
“Proposed by Kaidy Tan, Fukien Teachers’ University, Foochow, Fukien, China.
An isosceles triangle has vertex A and base BC. Through a point F on AB, a perpendicular to AB is drawn to meet AC in E and BC produced in D. Prove synthetically that
Area of AFE = 2 Area of CDE if and only if AF = CD.”
See the Linked Triangles Problem
(Update 2/22/2023) Alternative Solution
The indefatigable Oscar Rojas has come up with an alternative solution to the problem. It takes a bit of study to verify.