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| Four Disks (Posted on 2005-02-02) |
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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.
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Submitted by David Shin
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Rating: 3.0000 (4 votes)
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Solution:
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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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