User:Number77

File:Chordof.jpg

First focus on the center square of the two overlapping circles.

File: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 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,

$\ M=\ {\sqrt {2}}r-r$

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

So,

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

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

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

$\ r/C=\ r/(r-{\sqrt {2}}/2r)=2+{\sqrt {2}}$