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20090314For this problem refer to Ladder vs Alley problem.
Inverse Circle
What is it that I am calling an “inverse circle” ? An example will explain. Suppose you, the mathematician, have a given radius and a given point. You could easily draw a circle of that given radius using the given point as the circles center. But let’s say you want to draw a circle of a given radius with the point positioned on the circle’s circumference. That is what I am referring to as the “inverse of the circle”. It is probably not mathematically correct to call it that, but it will work to demonstrate an idea in geometry.
It is a simple observation to find this inverse circle. Take the given point and draw an arc of twice the radius of the given radius. On this arc’s circumference, take the original, “given” radius and draw a circle. The point will be positioned on the new circle’s circumference.
Now for some examples of how this knowledge may be useful. As you may recall from the ladder in the alley problem, the angle of ladders was determined by utilizing circle constructions. Drawing the inverse circle of the last paragraph is just one of the many possibilities. There are infinitely many positions on the circle’s circumference that the given point could reside. So in order for the circle to be useful we must know certain values that are given by the problem. For instance in the ladder problem we knew the length of both ladders (3 meters and 5 meters respectively) and we knew the ladders crossed 1 meter above the ground. Now do you see the possibilities and applications of this observation!
Below are some CAD drawings of the inverse circle. Pay attention to the loops of the many circles. It is much like a spiral graph. There is much to be learned here.
May the Creative Force be with You