Incenter and Conway's Circle

Date: 12/17/2002 at 18:13:32
From: Bonnie
Subject: Triangle and circle

In a triangle, the bisectors of the angles intersect at a point in the 
interior of the circle. If I use this point as a center to draw a 
circle, what is the relation of this circle to the triangle?  

Please help me with the answer. Thank you.

Date: 12/18/2002 at 03:54:07
From: Doctor Floor
Subject: Re: Triangle and circle

Hi, Bonnie,

Thanks for your question.

This question uses the following: The bisector of an angle is the 
locus of points that have equal distances to the legs of the angle. 
For instance, any point on the angle bisector of angle A is as far 
from AB as from AC.

The point of intersection is called the incenter. This point is on 
all three of the angle bisectors, and thus its distances to the sides 
of ABC are all equal. If we take this distance as the radius of a 
circle and I as center, then the circle is tangent to the three 
sides. This circle is called the incircle.

This may be the only circle that the questioner wants to hear about. 
But we can say a lot more about circles with I as center. If we take 
as radius a distance greater then the radius of the incircle, then 
the circle intercepts three chords on the (if necessary extended) 
sidelines of ABC:

  
The intercepted chords EF, GH, JK are congruent. This can be seen 
from the fact that the triangles IEF, IGH, and IJK are isosceles 
triangles with congruent top angle sides and congruent altitudes from 
their top vertices. So the triangles are themselves congruent.

So circles with I as center intercept congruent chords on the 
sidelines of ABC.

There is one especially well-known circle among these circles. This 
circle intercepts chords that have the perimeter of ABC as length. It 
is called Conway's circle:

  

The indicated lengths a, b, and c suggest how you can construct this 
circle.

If you have more questions, just write back.

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