Inscribed, Circumscribed Circles

Date: 04/25/2003 at 09:44:49
From: Hady
Subject: Equation of a circle

Given three general points in a plane of coordinates, (a,b), (c,d), 
and (e,f), what are the equations of the circles circumscribed about 
and inscribed within the triangle they form?

We need the general equation in order to generate a program 
concerning inserting the coordinates of any three points and getting 
the equations of their two circles.

Date: 04/25/2003 at 15:51:33
From: Doctor George
Subject: Re: Equation of a circle

Hi Hady,

Thanks for writing to Doctor Math.

When you are programming, it is often easiest to develop a way to 
compute the solution without directly solving equations. See if this 
helps.

Let's do the circumscribed circle first.

   1. Pick any two sides of the triangle.
   2. Construct the perpendicular bisecting line for each of those 
      sides.
   3. Find the intersection of those two lines. That point will be the 
      center point.
   4. The distance from the center to any vertex will be the radius. 
      All three distances should be the same.

Now let's do the inscribed circle.

   1. Pick any two vertices of the triangle.
   2. Construct the angle bisector for each of those angles.
   3. Find the intersection of those two lines. That point will be the 
      center point.
   4. The distance from the center to any side will be the radius. 
      All three distances should be the same.

Do you understand why these solutions work?

Once you have the center point and radius for each circle you can 
write down their equations.

If you need more help along the way please write again.

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