User:Number77

A circle can be placed so that its center is at the origin of an x y coordinate plane. A second circle is set on this coordinate plane with two of its sides touching the x and y axis. The radii are then drawn out every 90º for each circle. Draw a line segment connecting the center of the two circles. A smaller circle is then placed between the origin and the second circle.

In order to find the length of this diagonal line segment, the pythagorean theorum must be used.
 * $$\sqrt{r_{x}^2 + r_{y}^2} = \sqrt{2r^2} = chord\ length \,$$
 * $$r = radius \,$$

The height of the chord can be found by...
 * $$\ r - ({\sqrt{2r^2}}/2) = chord\ height = C    \,$$

The length of the segment from the origin to the second circle can be found by...
 * $$\sqrt{2r^2} - r = origin\ to\ circle_2\ segment = M \,$$

With this information known, we can relate them in the form of constants...


 * $$\ r/{(r - ({\sqrt{2r^2}}/2))} =\ r/C\ =\ 3.414213562\,$$


 * $$r/{(\sqrt{2r^2}}) - r =\ r/M\ =2.414213562 \,$$


 * $$(\sqrt{2r^2} - r) / (r - ({\sqrt{2r^2}}/2)) = M/C = 1.414213562 \,$$


 * $$(\sqrt{2r^2} - r)/r = M/r = 0.414213562\,$$


 * $$1.414213562 = \sqrt{2} \,$$

And the pattern can readily be seen by...
 * $$3.414213562 = 2 + \sqrt{2} \,$$
 * $$2.414213562 = 1 + \sqrt{2} \,$$
 * $$1.414213562 = 0 + \sqrt{2} \,$$
 * $$0.414213562 = -1 + \sqrt{2} \,$$

We can more easily see the reasons for these values as follows. First,


 * $$\ M =\ \sqrt{2} r - r $$
 * $$\ C =\ r - \sqrt{2}/2 r $$

So,


 * $$\ M/r =\ (\sqrt{2} r - r) / r = -1 + \sqrt{2} $$
 * $$\ M/C =\ (\sqrt{2} r - r)/(r - \sqrt{2}/2 r) = 0 + \sqrt{2} $$
 * $$\ r/M =\ r/(\sqrt{2} r - r) = 1/(\sqrt{2} - 1) = 1 + \sqrt{2} $$
 * $$\ r/C =\ r/(r - \sqrt{2}/2 r) = 2 + \sqrt{2} $$