Pascal in a Cyclic Quadrilateral: What is this about?
A Mathematical Droodle


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Explanation

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Copyright © 1996-2017 Alexander Bogomolny

A nice problem has been posted by Darij Grinberg at the Hyacinthos group:

  Given a cyclic quadrilateral ABCD with the circumcenter O. The perpendicular to BD through B meets the perpendicular to AC through C at E. The perpendicular to BD through D meets the perpendicular to AC through A at F. Finally, let X be the intersection of the lines AB and CD. Then, the points O, E, F, X are collinear.

An elegant solution was found by Jean-Pierre Ehrmann.


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The lines perpendicular to AC and BD form a parallelogram whose center coincides with the center O of the circle. Therefore E, F, and O are collinear. (In fact O is the midpoint of EF.)

In order to prove that X lies on the same line, construct the antipodes A' of A and D' of D on the circle. Note that Pascal's theorem is applicable to hexagon ABD'DCA'. The theorem says that the three points E, F, and X lie on Pascal's line of the hexagon.

Pascal and Brianchon Theorems

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Copyright © 1996-2017 Alexander Bogomolny

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