I was reading about the the four turtles/bugs math puzzle Four bugs are at the four corners of a square of side length D. They start walking at constant speed in an anticlockwise direction at all times directly towards the bug ahead of them. How far does each bug walk before they meet with each other?
and the a-ha answer given is that the turtles are always in the shape of a square, and since each one is always walking perpendicular to the direction of his pursuer's approach, he neither hinders nor helps the pursuer in reaching him. Therefore the time to the center is the same as the time it takes one turtle to walk across a side of the square.
Since each creature is running directly to its target, the paths of a chaser and its chasee are always at right angles. That is, the distance between the chaser and its chasee depends only on the movement of the chaser, not its chasee. Therefore, the distance that the chaser runs is the same as if the chasee did not run at all.
Could you please explain how this happens?