JEE Advance - Mathematics (1989 - No. 17)
Let $$ABC$$ be a triangle with $$AB = AC$$. If $$D$$ is the midpoint of $$BC, E$$ is the foot of the perpendicular drawn from $$D$$ to $$AC$$ and $$F$$ the mid-point of $$DE$$, prove that $$AF$$ is perpendicular to $$BE$$.
This problem requires vector algebra to solve efficiently.
Using coordinate geometry simplifies the proof significantly.
The problem can be solved by proving that the dot product of vectors AF and BE is zero.
Proving triangles congruent is the most straightforward approach.
This is an application of Ceva's Theorem.