Consider the following figure... The red lines are extended radii of
each circle, tangent to the other circle. Prove that the
chords cut off by these radii, the green lines, are equal in length. That
is, prove that x=y.

To begin the proof, we'll keep just one of the four tangents. Then, in
each circle we will draw an extra radius and a horizontal line. The left circle's
radius we'll call r_{1}, and the right circle has radius r_{2}. The distance between the
centers of the circles is D. Now consider these two triangles: one, mostly
purple,
is formed by the horizontal line connecting the centers of the circles, a radius
of the right circle, and a tangent to the right circle, colored purple.
The other one, mostly green, consists of the same purple tangent, a horizontal
line, and a vertical line. Since the two triangles are both right
triangles, and they have one other angle in common, they are similar.
Thus,

2r_{1}/D = x/r_{2},

so

x = 2r_{1}r_{2}/D

In the next diagram, We keep the line connecting the centers of the circles,
length D, just as it was. But now, we interchange the roles of the two
circles and their radii. Once again, we have similar triangles, so,

2r_{2}/D = y/r_{1},

so

y = 2r_{2}r_{1}/D

This completes the proof that x=y.

Here is the diagram again, with all the lines shown, so that you will be
convinced the "x" in figure 2 and the "y" in figure 3 are the same as the "x"
and "y" in figure 1:

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Summary of geometrical theorems

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