Sum of three segments

Problem

ABC is a equilateral triangle. P is a point inside the triangle. Draw three line segments PD, PE, and PF perpendicular to BC, CA, and AB respectively.
Prove PD+PE+PF is constant.

Applet

  1. Drag the red point indside the triangle.
  2. Check "Hint" check box. What the figure imply?
  3. Click the "Hint" check box again.
  4. Click "mode 2" check box. The point will be fixed on the side BC.
  5. Drag the red point on the side.
  6. Check "Hint" check box. What the figure imply?