The Broken Chord Theorem
Proof by Mariano Perez de la Cruz

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 by Mariano Perez de la Cruz

Proof

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

Copyright © 1996-2017 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 by Mariano Perez de la Cruz

By extending PM to its intersection with the circumference we get point S. A line from S through B intersects the AC extended in F.

∠MSA = ∠MSF since P bisects the original arc AB, implying AM = MF.

We are to prove that CF = CB.

This is so because, ∠CBF is suplementary of ∠CBS, and, since ACBS is an inscribed quadrilateral, ∠CAS = ∠CFB, both being suplementary to ∠CBS. Therefore CF = CB.

The Broken Chord Theorem

  1. The Broken Chord Theorem: Proof Close to Archimedes'
  2. The Broken Chord Theorem: proof by Gregg Patruno
  3. The Broken Chord Theorem by Paper Folding
  4. The Broken Chord Theorem: proof by Stuart Anderson
  5. The Broken Chord Theorem: proof by Bui Quang Tuan
  6. The Broken Chord Theorem: proof by Mariano Perez de la Cruz
  7. Pythagoras' from the Star of David
  8. Pythagoras' from Broken Chords
  9. Extremal Problem in a Circular Segment

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

Copyright © 1996-2017 Alexander Bogomolny

 61200884

Search by google: