Concyclic Points in Bride's Chair: What Is This About?
A Mathematical Droodle

 

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The applet attempts to illustrate a property of the configuration known as the Bride's Chair.

 

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It is well known that lines AK and CD are perpendicular as are the lines CE and BF. Let U denote the intersection of AK and CD and V the intersection of CE and BF. Let C(ABDE) be the circle circumscribed around square ABDE. Then both U and V lie on that circle.

The proof is simple. The diagonals AD and BE serve diameters of the circle C(ABDE). ∠BUE is right and is subtended by the diameter BE. By the converse of a property of inscribed angles, U lies on the circle C(ABDE). The same reasoning applies to ∠AVD, so that V, too, is on the circle.

A useful consequence of this result is the identity CE·CU = CD·CV which follows by the Intersecting Secants theorem.


Related material
Read more...

  • Bride's Chair
  • Bride's Chair: an Interactive Gizmo
  • Desargues in the Bride's Chair(with Pythagoras)
  • Joseph Keech in Bride's Chair
  • A Square in the Chair

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