Hinge Theorem

Date: 12/12/2000 at 18:54:06
From: Gina
Subject: Proofs in geometry

How would you write a proof for the Hinge theorem?

Date: 12/13/2000 at 05:10:36
From: Doctor Floor
Subject: Re: Proofs in geometry

Hi, Gina,

Thanks for writing.

Hinge Theorem:

If of triangle ABC and A'B'C' sides AB = A'B' and AC =  A'C', and 
the included angle at A is larger than the included angle at A*, then 
BC > B'C'.

Proof:
                    A              A'
                   /|\            /|
                  / | \          / | 
                 /  |  \        /  |
                /   |   \    B'/   |
               B    | X  \C        |C'
                    D

We construct AD such that AD = A'C' = AC and angle BAD = angle B'A'C'.
Triangles ABD and A'B'C' are congruent. Therefore BD = B'C'.

Let X be the point where the angle bisector of angle DAC meets BC.
From the congruent triangles AXC and AXD (SAS) we have that XD = XC.
Now, by the triangle inequality we have that BX + XD > BD, so 
BX + XC > BD, and consequently BC > BD = B'C'.

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/