[N.B. This applet assumes that you have already convinced yourself that the sum of the lengths from any point inside an equilateral triangle to the sides of the triangle is constant!]
The GOLD point in the left panel is chosen at random inside the equilateral triangle. The red, blue and green segments are lines drawn from the GOLD point and perpendicular to each of the sides. Their lengths vary in size as you move the GOLD point from place to place inside the triangle. However, the sum of their lengths is constant.
What is the sum equal to? Why? Can you prove it?
[You can change the size of the equilateral triangle by dragging the BLACK dots.]
Would a similar thing be true in a square? Why or why not?
What about other regular polygons with an odd number of sides? with an even number of sides?
Under what circumstances can the red, green and blue lengths can form a triangle?
Can you prove your conjecture?