Tangents to two circles
Chris Sangwin
1 Introduction
The problem:
Given two circles, find lines tangent to both.
2 Tangents to a circle
The key to this problem is to the ruler and compass construction of the tangent
to a circle through a point. The complete construction is shown below.
Figure 1: Constructing the tangent to a circle
The key points are
- Construct the midpoint of the point and centre of circle
- Draw the circle through the midpoint and centre of circle
- The intersection of the two circles are the points of tangency
3 The problem to solve
The GeoGebra worksheet below explains the problem to be solved.
We place a point E on the line between the centres of the circles. Next
construct the points of tangency to both circles using the above construction.
We connect the points of tangency, where these intersect we have the point L.
The problem is to find the position of E so that E and L coincide.
Figure 2: The problem to be solved
Notice there are a number of cases. The first is as shown above.
The second can be illustrated by dragging E so that it
no longer lies between A and B and the third
occurs when both circles are the same radius.
Similar triangles AKL and BIL is the key.
The solution is an algebraic one, although a pure ruler and compass construction is
possible, but just more complex and obscures the details. To see this see the separate page
here.
4 The solution
Based on this we have the following construction.
Figure 3: The solution
This is encapsulated in a GeoGebra tool 
You can test the tool with the worksheet below.
Figure 4: Testing the tool
5 A tangent theorem
Take any three circles, of different radii.
In pairs, find the intersection of the external mutual tangent lines.
Prove the three intersection points are colinear.
A proof is given here.