Let ABC be a right triangle with A the right angle and let D be the mid-point of side AB.
If E is the foot of the perpendicular from A to CD and F is the mid-point of CE,
prove that BE is perpendicular to AF.
Draw triangle ABC with the other lines as described. Rotate triangle 90 degrees clockwise around (at) point C and redraw the same triangle and lines. Clearly line BE in the original triangle on the left is parallel to line AF in the second triangle on the right. Accounting for the 90 degree rotation, the relationship of BE to AF is therefore 90 degrees (in either triangle)!
But Daniel did a nice job with the actual math involved!
Simpler approach!
