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Four Tangent Circles (Posted on 2003-10-20) Difficulty: 4 of 5
Four circles are tangent to each other and to a triangle that surrounds them, as shown in the figure. Each of the three circles in the corners have a radius of 1 cm. What is the radius of the circle in the middle?

No Solution Yet Submitted by Bryan    
Rating: 4.1429 (7 votes)

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re: solution | Comment 5 of 13 |
(In reply to solution by SilverKnight)

Your equation 2*R^2 - 12*R + (2*R+2)(sqrt(R^2+2*R)) + 2 = 0
reduces to cubic equation 16*R^3 - 33*R^2 + 14*R - 1 = 0.

That equation has three positive roots: R=1.51067547264, R=0.462340147657, and R=0.0894843796967, but since the radius must be large enough to touch the triangle, the only solution is R=1.51067547264.
Edited on October 21, 2003, 9:32 am
  Posted by Brian Smith on 2003-10-21 09:30:20

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