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 Four Disks (Posted on 2005-02-02)
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: a+b+c+d=180 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.

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 Subject Author Date Added Observation Bractals 2005-02-04 23:03:42 re(2): Initial Thoughts Jer 2005-02-02 20:12:21 re: Initial Thoughts Hugo 2005-02-02 18:49:04 solution Jer 2005-02-02 17:32:27 Initial Thoughts nikki 2005-02-02 14:26:50
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