Finding the Incenter of a Triangle

Date: 10/08/2006 at 17:58:54
From: Eli
Subject: Finding the Incenter of a Triangle

Hi, I am wondering what is the equation to find the incenter of a 
triangle?  I do not understand the Cartesian Coordinates of the 
incenter.

Date: 10/08/2006 at 18:52:15
From: Doctor Philip
Subject: Re: Finding the Incenter of a Triangle

Hi Eli,

One way to find the incenter of a triangle in the Cartesian plane is
to use the fact that it is equidistant from all three sides.

If you find the equations of all three sides of the triangle, and put
them into standard form, you can then use the formula for the distance
between a point and a line to find the incenter.

For example, if two of the sides are

  a1(x) + b1(y) = c1
  a2(x) + b2(y) = c2             Note: the numbers are subscripts

then a point is on the bisector of the angle formed by these two lines is

  |a1(x) + b1(y) - c1|     |a2(x) + b2(y) - c2|
  --------------------  =  --------------------
   sqrt((a1)^2+(b1)^2)      sqrt((a2)^2+(b2)^2)

You can write a similar equation using two other sides of the
triangle.  Finding the intersection between the two bisectors yields
the incenter.

Please write back if you have any more questions on this topic.

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

Date: 10/09/2006 at 11:23:26
From: Eli
Subject: Thank you (Finding the Incenter of a Triangle)

Thanks for your help.  It helped me understand incenters in a very
easy way.