Geometry Problem 1121

Casey's Theorem. Generalized Ptolemy's Theorem, Art, Poster

See also:
Art of problem 1439, Generative Sketching.

Proposition

In the figure below, the circles A,B,C,D are tangent to the circle O; a, b, c, d, x, y are the length of the exterior common tangent to circles A and B, B and C, C and D, D and A, A and C, and B and D respectively. Prove the generalized Ptolemy's theorem: \(x \cdot y = a \cdot c + b\cdot d\).

Figure 1

 

Figure 2

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