Four disks are arranged in a plane such that each is externally tangent to two others. Prove that the four points of tangency lie on a circle.

(In reply to Initial Thoughts by nikki)

First of all: Nice work Jer!

Nikki, you where almost there.
I continued on your posting and imagined a point S near the place where the center of the circle we are looking for should be.

Connect S to A, to B and to M
Writing down all the equations of the sum of the angles in the triangles and the quadrilaterals, you end up with a number of equations that end in angle AMS = BMS = 90°.  From there on it is easy to see that AS = BS.
When doing your reasoning + the above for the four pairs of tangent circles, you can prove that all the lengths from S to the tangent points are the same, and must be lying on a circle.
But Jer's work is more elegant then ours :(  

Thanks David, another puzzle I had a lot of fun with.

Posted by Hugo on 2005-02-02 18:49:04