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The Real Cyclic Quad (Posted on 2014-08-23) Difficulty: 3 of 5
About a set of four concurrent circles (circles passing through a one common point) of same radius r,four of the common tangents are drawn to determine the circumscribing quadrilateral ABCD.Prove that ABCD itself is a cyclic quadrilateral.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 1.5000 (2 votes)

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re: Part of it. Comment 2 of 2 |
(In reply to Part of it. by broll)

The corresponding angles of ABCD and O1O2O3O4 are equal, but the lengths of the sides are not proportional. Therefore, the quads are not similar.

Since O1O2O3O4 is cyclic, opposite angles are supplementary. Hence, the corresponding angles of ABCD are supplementary. Therefore, ABCD is cyclic.

Also, when labeling the centers O1, O2, O3, and O4 care should be taken that  O1O2O3O4 is convex.

Edited on August 23, 2014, 12:17 pm
  Posted by Bractals on 2014-08-23 12:09:38

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