Line Tangent to Two Circles

Date: 07/01/99 at 22:11:40
From: RC Cruz
Subject: Geometry

How do you create a line tangent to two different sized circles?

Date: 07/02/99 at 02:28:47
From: Doctor Floor
Subject: Re: Geometry

Hello and thanks for your question!

Let me use (C1) and (C2) for the two different-sized circles with 
centers C1 and C2. Draw the line l = C1C2, and the perpendiculars p1 
and p2 to that line through C1 and C2 respectively.

The line p1 meets (C1) in two points P1 and Q1, and P2 and Q2 are
found on (C2) similarly.

The lines P1P2, Q1Q2, P1Q2, and P2Q1 intersect l in two points: 
A and B. It is only useful to consider these points if they are 
outside (C1) and (C2), as both will be when the two circles do not 
intersect and are apart from each other; one of them will be outside 
when one circle is not totally inside the other one. 

A and B are the starting points for possible tangents to both circles. 
If a tangent starts from, for instance, A, then the reflection of that 
tangent through l is of course a tangent too, and thus two (or zero) 
tangents start from one point.

Now you can construct the tangents from, for instance, A to (C1) in 
the following way: 

Draw the circle having AC1 as diameter, and find the intersection 
points of this circle with (C1), say X1 and X2. The lines AX1 and AX2 
are the two tangents we look for.

When two circles are do not intersect, you will find a total of four 
common tangents to those circles.

If you need more help, just write us back.

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