In a triangle ABC, AB = AC. D is the mid-point of BC, E is the foot of the perpendicular drawn from D to AC, and F is the mid-point of DE.

Question

Let ABC be a triangle with AB=AC. If D is the mid point of BC, the foot of perpendicular drawn from D to AC and F is the mid point of DE. Prove, without using coordinate geometry, that AF is perpendicular to BE.

Answer 1

Take BC as the x-axis and D as the origin; then clearly DA is y-axis.

Take coordinates of B as (-a, 0) and C as (a, 0). Also, take coordinates of A as (0, b). Equation of AC is