Congruent Triangles

Date: 6/26/96 at 8:21:52
From: Anonymous
Subject: Congruent Triangles

Dr. Math,

If two triangles have the same area AND the same perimeter, must they 
be congruent? If the answer is no, could you please provide me with a 
counterexample?  Thanks a lot.

I think the answer is no, but I have been unable to find a 
counterexample

Doug Lewis

Date: 6/26/96 at 14:57:3
From: Doctor Ceeks
Subject: Re: Congruent Triangles

Heron's formula gives:

A^2 = s(s-a)(s-b)(s-c), where s is the semi-perimeter (a+b+c)/2.

The question is: do there exist a,b,c and a',b',c' which satisfy the 
triangle inequalities for which a+b+c = a'+b'+c' and 
(s-a)(s-b)(s-c) = (s-a')(s-b')(s-c') where we define 2s = a+b+c?

Because the set of a,b,c which satisfies the triangle inequality forms
a 3-dimensional space, and because fixing the area and the perimeter
eliminate a total of two degrees of freedom, one can expect the answer
to be yes.

And in fact, a little playing around gives:

5,12,13 and 9, (21-root(41))/2, (21+root(41))/2

as the sides of two triangles with the same area and perimeter,
and in fact the area and perimeter are both 30.

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/