The GOLD point is chosen at random inside the equilateral triangle. [You can change the size of the equilateral triangle by dragging the BLACK dots.] 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. Why? How is the sum of the lengths related to the size of the triangle? Why? Can you prove it?
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?
If the GOLD point is dragged outside the triangle the relationship still holds if the distance from the GOLD point to the triangle is counted as negative! Why?