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Geometry: Minimum Distances and Circles

From: Felix Sheng-Ho Chang
Date: 03 Nov 1994 16:45:00 PDT
Subject: A Difficult Problem (I Need HELP!)

Hi. I'm Felix, a high school student in Burnaby South, Burnaby, BC, 
Canada. Here is a problem that I have had difficulty with for a long 

Given an arbitrary circle, inside the circle, choose two arbitrary
points A & B, is there a way to find a point C ON THE CIRCLE,
such that the sum of the length AB and the length BC is a minimum?

(Using compass and ruler ONLY)

Felix Sheng-Ho Chang
Burnaby South Secondary School
Burnaby, B.C., Canada.
The Opinions of the Students Do Not Necessarily Reflect That of the 

From: Dr. Ethan
Date: Thu, 3 Nov 1994 19:57:58 -0500 (EST)

     Well, with the help of my colleague Dr. Kenneth "Wildman" 
Williams we think that we have found a solution.

     Here is a hint, Find the center of the circle. The use it and B to 
do something neat.  If you need more than this write back.
        Ethan Doctor on CAll

From: "John Conway"
Date: Sun, 6 Nov 94 18:49:25 EST

     I just tried to answer F CHANG's interesting question "at the
keyboard", and had to give up when I got into some complicated
formulae.   I'm going to try to get back to it (I'm pretty sure
I can answer it), but the following remarks may be helpful.

     I believe you meant to ask for the point C on the circle that
makes  AC + BC minimal (you said something else).  Anyway, that's
the question I'm thinking about.  Well, the locus of the C for
which AC + BC takes a given value is an ellipse with foci A and B,
so we want the smallest such ellipse that surrounds the circle,
which will touch it at C.

   I plan to work out the equation whose roots correspond to
all the ellipses with foci A and B that touch the circle.  If
this turns out to be in general an irreducible equation of degree
higher thanm 2, I'll probably be able to prove there's no
such way.  That's what I guess will turn out to be true.

          John Conway
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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