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Three Circles (Posted on 2004-07-28) Difficulty: 4 of 5
Three circles of radius 6, 7, and 8 are externally tangent to each other. There exists a smaller circle tangent to all three (in the space created between the three original circles).

What is the radius of this smallest circle?

No Solution Yet Submitted by ThoughtProvoker    
Rating: 3.0000 (3 votes)

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re(2): Inversion | Comment 6 of 7 |
(In reply to re: Inversion by Thalamus)

Inversion about ( or through ) a circle with center O and radius r is a mapping of the plane such that for every point P and its image P' we have |OP||OP'| = r^2. O, P, and P' are collinear with O not between P and P'. Clearly, a point on the circle is mapped into itself and O and the point at infinity are images of each other. What's nice about the mapping is that 1) a line through O is mapped into itself, 2) a line not through O is mapped into a circle through O with center on the perpendicular from O to the line, 3) a circle not through O is mapped into a circle with center on the line through O and the center of the circle being mapped, 4) a circle through O is mapped into a line which is perpendicular to the line through O and the center of the circle being mapped, 5) angles between objects ( lines or circles ) are preserved, 6) points of tangency are prserved, etc. What we do is take a difficult problem, like constructing the Soddy circles, and invert it about a circle with a center and radius which will map it into a simpler problem. We solve the simpler problem - then invert it back using the same inversion. 
  Posted by Bractals on 2004-07-30 20:42:26

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