Diameter of a Circle Circumscribed Around a Triangle

Date: 05/13/98 at 05:01:03
From: Eric
Subject: Circumscribed Triangle

The sides of a triangle are 25, 39, 40. Find the diameter of the 
circumscribed circle.
 
Please help. I don't even know how to begin, let alone finish it! 
Thank you very much!

Date: 05/14/98 at 17:43:40
From: Doctor Bob
Subject: Re: Circumscribed Triangle

Hello Eric,

This appears to me to be a tough one! I am going to be very interested 
in any other ideas about it.

I can see a way to get an answer by analytic methods. By this, I mean 
that we could draw the triangle in a rectangular coordinate system, 
then determine coordinates for each vertex and for the center of the 
circle.

Place one end of the 40 unit-long side at (0,0) and the other at 
(40,0). This means the center of the circumscribed circle will be at 
(20,b). If we can only find b then, the Pythagorean Theorem would 
carry the day! I suggest that finding the coordinates of the third 
vertex would be a big help. We should be able to do that since we 
could assume that this vertex is 25 units from (0,0) and 39 units from 
(40,0). (Assume the triangle is placed above the x-axis, to keep 
coordinates positive.)

If you are comfortable working with coordinate geometry, the method 
above will work out. The numbers are even selected so that the algebra 
works out nicely. I have left out quite a bit of detail that I hope 
you can supply. Please write back if you need more help with the 
solution.

I would love to see a good synthetic (i.e., coordinate-free) approach.

-Doctor Bob, The Math Forum
 http://mathforum.org/dr.math/