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

 

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

A few words

|Activities| |Contact| |Front page| |Contents| |Geometry| |Store|

Copyright © 1996-2017 Alexander Bogomolny

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

 

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

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
  • |Activities| |Contact| |Front page| |Contents| |Geometry| |Store|

    Copyright © 1996-2017 Alexander Bogomolny

     61257247

    Search by google: