Isosceles Triangle Proof

Date: 05/14/2006 at 13:29:17
From: Jeroen
Subject: Isosceles triangle proof.

Given triangle ABC, with D on BC and AD bisecting angle A.  The center
of the circle circumscribing ABC is the same point as the center of
the circle inscribed in ADC.  Prove that ABC is a isosceles triangle.

I think it's difficult because I can't get a drawing of it!  I can't 
get the centers of the 2 circles drawn at one place. 

I know that the circumcenter is the center of a circle passing through
the three vertices of the triangle and the intersection of the angle
bisectors finds the center of the incircle.  Can you help me?

Date: 05/15/2006 at 12:02:04
From: Doctor Peterson
Subject: Re: Isosceles triangle proof.

Hi, Jeroen.

This is a very nice problem, and it raises some interesting issues!

My first comment is that you don't need to be able to make an 
accurate drawing in order to do a proof; often I just make a drawing 
that illustrates the relationships (e.g. this point is on that line) 
and just make marks on it to indicate that certain segments are 
SUPPOSED to be congruent, and so on.  In fact, sometimes you want to 
deliberately make an inaccurate drawing; for example, if you are not 
assuming that a triangle is isosceles, but need to prove it, you may 
want to draw a scalene triangle so that you don't accidentally 
conclude something based on that unjustified assumption.  When you do 
so, the drawing will not really meet the premises of the theorem, 
but you can pretend.

But it IS helpful to have a drawing that is reasonably CLOSE, and 
sometimes even that can be hard!  It will help to take the conclusion 
into account when you make a drawing; since you know (if it's true!) 
that the triangle has to be isosceles, it doesn't make sense to try 
drawing it with any old scalene triangle.  In this case, I just 
happened to draw a triangle that worked on my first try; since they 
don't tell you which two sides will turn out to be congruent, I 
might have had to make three tries.  But when I tried to make 
drawings with different angles (in order to get a better feel for 
how the problem works, and confirm that it always does), I found 
that doing exactly the same work DIDN'T always work -- it turns out 
that this theorem's converse is not true, so it isn't just ANY 
isosceles triangle that you can draw.

Another thing that can help when you want to draw a figure for a 
problem like this is to work backwards -- which can also help a lot 
in the exploratory phase of developing the proof.  In this example, 
rather than just draw a triangle to start with, you can first draw 
the incircle and circumcircle, with the same center, then create a 
triangle that will fit them: circumscribe a triangle around the 
incircle (with two vertices on the circumcircle) and extend side CD 
to find vertex B on the circumcircle.  You may find that angle A is 
not bisected, but you at least have a figure that meets most of the 
requirements, and you can just label the "bisected" angle to claim 
that it is!

In the process of drawing this, you may have discovered some of the 
interesting relationships that will lead to a proof.  Here's a hint: 
draw in the radii from the center to vertices A, B, and C, mark them 
as being congruent, and look for some provably isosceles triangles. 
Then label a bunch of little angles that are all congruent; I called 
them all x, and labeled some bigger angles as multiples of x.  You'll 
find that x has to have one specific value, so that we could conclude
not only that ABC is isosceles, but that it is a specific (and even
somewhat familiar) triangle.

If you have any further questions, feel free to write back.


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

Date: 05/19/2006 at 16:22:13
From: Jeroen
Subject: Isosceles triangle proof.

I thank you very much for giving me a tip for this problem (I've found
a nice proof now).