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 Math Help > Geometry > Circles, Conic Sections > Mutual Radial Tangents Puzzle

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 r1, and the right circle has radius r2.  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,

2r1/D = x/r2,

so

x = 2r1r2/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,

2r2/D = y/r1,

so

y = 2r2r1/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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The webmaster and author of this Math Help site is Graeme McRae.