User:Number77

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Image:Chordof.jpg
First focus on the center square of the two overlapping circles.
Image:Chorddouble.jpg
Another view of the math

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 90ยบ 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}