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        riddles >> medium >> Two Triangles
        
(Message started by: THUDandBLUNDER on Apr 6^th, 2004, 9:33am)

Title: Two Triangles
Post by THUDandBLUNDER on Apr 6^th, 2004, 9:33am

Let ABC be a primitive right triangle with integer sides and integer area.
Let DEF be an isosceles triangle with integer sides and integer area.

If ABC and DEF have the same perimeter and the same area, find the lengths of the sides of ABC and of DEF.

Title: Re: Two Triangles
Post by Sir Col on Apr 6^th, 2004, 4:59pm

I don't have a solution, but I've made a start (albeit fairly obvious)...

::[hide]
Let the sides of a right angle triangle are given by x=2mn, y=m²–n², and z=m²+n² (hypotenuse).

P=2m(m+n), and A=mn(m+n)(m–n); so A=P(m–n)/2.

As the right angle triangle is a primitive, one of m and n must be odd and the other even, otherwise all the sides would be even.

For the isosceles triangle to have integer sides and integer area, the base*height/2 must be integer, which means that it could be two Pythagorean right angle triangles back-to-back; although this is not necessary: the base of each right angle triangle could be a multiple of 0.5 and the height a multiple of 2.

Assuming that the three sides of the right angle triangle are given by, a=2jk, b=j²–k², and c=j²+k² (hypotenuse).

Depending on orientation of the right angle triangles, the sides of the isosceles will be c-b-c or c-a-c.
P₁=2(j²+k²)+2(j²–k²)=4j²
P₂=2(j²+k²)+2*2jk=2(j+k)²

But whichever orientation, A=2jk(j+k)(j–k).

From here, I'm unable to establish anything else.
[/hide]::

Title: Re: Two Triangles
Post by SWF on Apr 7^th, 2004, 5:06pm

Answer:
[hide]Using expressions similar to Sir Col's for Pythagorean triples, also find the area in terms of perimeter with Heron's formula. For the isosceles triangle with base q, and leg r:
16*A*A=p*(p-2r)*(p-2r)*(p-2q)
This implies p*(p-2q) is a perfect square,...

Right triangle: a=135, b=352, c=377
Isosceles: q=132 r=366
Perimeter=864 Area=23760
[/hide].

Title: Re: Two Triangles
Post by NickMcG on Apr 7^th, 2004, 5:15pm

This was a recent "IBM Ponder This" problem. Amazingly the solution is unique.
See http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/solutions/February2004.html/$FILE/Feb2004_dima.pdf

Title: Re: Two Triangles
Post by Barukh on Apr 8^th, 2004, 5:51am

A very nice problem :D! I just wanted to post my solution, but saw SWF and NickMcG came first ;D

What about the following variation:

Find a pair of primitive isosceles Heronian triangles with the same perimeter and area.

Does a triple of such triangles exist?