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Chord Triangle Constant (Posted on 2015-03-21) Difficulty: 4 of 5

Let Γ be a circle with center O and radius r. Let P be a point inside Γ
( different from O ) with |OP| = p.

a) Prove there exists a point A outside Γ such that for all chords BC
of Γ through P the quantity (b+c)/a is constant ( where a, b, and c
are the side lengths of ΔABC ).

b) What is the constant in terms of p and r?

c) Prove that the point A is unique.

See The Solution Submitted by Bractals    
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Comments: ( Back to comment list | You must be logged in to post comments.)
re: Possible Solution Comment 4 of 4 |
(In reply to Possible Solution by Harry)

Liked your construction of the circle through points B, C, and O to find point A and the value of the constant (b+c)/a for that point A.

The first thing I thought of when I saw the problem was "point P inside Gamma and point A outside Gamma" and that P and A should be inverses with respect to Gamma.

You lost me about half way through the uniqueness part.

I used Geometer's Sketchpad to construct an ellipse.
  1) Constructed a fixed length segment B'C'.
  2) Constructed point A' such that triangles ABC and A'B'C'
      are directly similar.
  3) A' then traces out the ellipse as B traces out Gamma.

Edited on March 25, 2015, 10:23 pm
  Posted by Bractals on 2015-03-25 22:18:18

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