Tangent Circle Construction

Date: 12/02/96 at 23:48:58
From: Jimmy Midland
Subject: Euclidean geometry

Given a circle and two points inside of the circle, construct another 
circle which passes through the given points and is tangent to the 
given circle.

Date: 12/05/96 at 10:13:10
From: Doctor Yiu
Subject: Re: Euclidean geometry

Dear Jimmy,

This is one of the famous problems in geometric construction.  Here is 
one construction (without proof). If you would like an explanation of 
the basic principle behind the construction, please write back.  

I: First consider the case when A and B are EQUIDISTANT from the 
center O of the given circle. 

Draw the perpendicular bisector of AB to meet the given circle at two 
point P and Q.

The two circles ABP and ABQ are each tangent to the given circle.

II: Now, suppose that A and B are NOT equidistant from the center O of 
the given circle.

(1) Draw the perpendicular bisector of AB to meet the given circle at 
a point P. (There are two such intersections. Choose either of them).

(2) With P as the center, draw a circle through A (it also passes 
through B), and mark its intersections with the given circle as 
C and D.

(3) The lines AB and CD meet at a point K (since A and B are not 
equidistant from the center of the given circle).

(4) Suppose A is BETWEEN K and B. (If not, simply change the labels of
the points A and B).  Draw a circle with KB as  DIAMETER, and then
through A draw the perpendicular to KB to meet this circle at X. 

(5) Draw the circle with center K passing through the point X.  Mark 
the intersections of this circle with the GIVEN circle as M and N.

(6) Mark the intersection of OM and the perpendicular bisector of AB. 
With this as center, draw a circle passing through M.  This circle 
is tangent to the given circle and passes through the given points 
A and B.

(7) Repeat (6) with the line ON (instead of OM). This gives a SECOND 
circle meeting the same requirement.

Here is a diagram illustrating these steps: 


         


 and a sketch made with The Geometer's Sketchpad is at:
  
  http://mathforum.org/dr.math/sketches/tangent_circle.gsp      

To open the sketch, you'll need at least a demo version of the 
Geometer's Sketchpad, which you can obtain at:

 http://mathforum.org/dynamic.html   

-Doctor Yiu,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math