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Four Disks (Posted on 2005-02-02) Difficulty: 2 of 5
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.

  Submitted by David Shin    
Rating: 3.0000 (4 votes)
Solution: (Hide)
This solution is due to Jer:

Call the centers of the disks A, B, C, D, and let the four angle measures of quadrilateral ABCD be 2a, 2b, 2c, 2d. Since the angles of a quadrilateral sum to 360, we have:


Now, connect the four points of tangency. This forms four isoceles triangles with verticies A, B, C, D and also a new quadrilateral. The base angles of the isoceles triangles are 90-a, 90-b, 90-c, 90-d, respectively.

The new quadrilateral therefore has angles a+b, b+c, c+d, d+a.

Both pairs of opposite angles of this quadrilateral sum to a+b+c+d=180.

This is a sufficient condition for a quadrilateral to be inscribed in a circle. Therefore, the points of tangency lie on a circle.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some ThoughtsAdded ObservationBractals2005-02-04 23:03:42
re(2): Initial ThoughtsJer2005-02-02 20:12:21
re: Initial ThoughtsHugo2005-02-02 18:49:04
solutionJer2005-02-02 17:32:27
Some ThoughtsInitial Thoughtsnikki2005-02-02 14:26:50
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