Begin with one BLACK circle, and inscribe 4 identical, interior, cyclically tangent, CYAN circles, such that they are also tangent to the black circle.

In each of the four cyan circles, inscribe 4 RED interior circles in exactly the same way.

Finally, inscribe a small GREEN circle in the center of the original black circle, tangent to the cyan circles.

Which is larger, this small green circle, or one of the small red circles?

let's say the red circles have diameter x. then a square whose vertices are the centers of four red circles (within one blue circle) has sides x and a diagonal x√2.

thus the blue circle has a diameter x√2 + x. then the square whose vertices are the centers of the blue circles has sides x√2 + x and a diagonal √2(x√2 + x) or 2x + x√2.

that diagonal consists of two blue circles' radii and the diameter of the green circle. subtract the blue circle diameter from the diagonal to find the diameter of the green.

(2x + x√2) - (x√2 + x) = x

the green and red circles have the same diameter, in this case, the wonderful x

Posted by rixar on 2004-06-16 22:16:29