Associated Topics || Dr. Math Home || Search Dr. Math

### 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:

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
```
Associated Topics:
High School Conic Sections/Circles
High School Constructions
High School Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search