The Broken Chord Theorem
Proof by Bui Quang Tuan

On the circumcircle of triangle ABC, point P is the midpoint of the arc ACB. PM is perpendicular to the longest of AC or BC. Prove that M divides the broken line ACB in half.
broken chord theorem

Proof

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Copyright © 1996-2018 Alexander Bogomolny

On the circumcircle of triangle ABC, point P is the midpoint of the arc ACB. PM is perpendicular to the longest of AC or BC. Prove that M divides the broken line ACB in half.
broken chord theorem

Let O denote the circumcenter of ΔABC. P' = symmetry of P in O. Because P is midpoint of arc ACB so PP' is perpendicular bisector of AB and P, C are on one side with respect to AB.

The perpendicular from P' to AC cuts AC at M' and cuts the circumcircle (O) again at M''. Two angles CAB and M''P'P are equal because their side lines are respectively perpendicular. Hence their subtending chords are also equal:

CB = M''P = M'M.

By symmetry of circumcircle, CM = AM'. From these:

AM = AM' + M'M = MC + CB

And the proof is complete.

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

Copyright © 1996-2018 Alexander Bogomolny
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