You are asked to help design a Three Lock Box. Your job is to decide the locations of the three locks. The door is a unit square. To operate properly, each lock must be the same distance from the nearest door edge and also that same distance from each of the other two locks.
As the three locks must be the same distance from each other, they form the vertices of an equilateral triangle. The simplest way to place this is equidistant from the left and right door edges. If the base of the equilateral triangle is horizontal, each side, including the base, is 1/3 the width of the door, which is 1, so it is just 1/3.
The topmost lock is then positioned so that it is 1/3 unit from the top edge of the door.
As a check, the height of the triangle is sqrt(3)/6 and the base is .37799... from the bottom edge of the door, and so that is not the nearest edge to any of these and so doesn't have to match the distance.
The triangle could be tilted a little and made a little larger and brought down a little, and such a scheme would also work, until it was tilted so much that one vertex would be closer to the bottom edge than to the top.
Posted by Charlie
on 2010-07-19 16:13:53