How to construct (draw) the two internal tangents to two given circles

This page shows how to draw one of the two possible internal tangents common to two given circles with compass and straightedge or ruler. This construction assumes you are already familiar with Constructing the Perpendicular Bisector of a Line Segment.

How it works

The figure below is the final construction with the line PJ added.

The construction has three main steps:

The circle OJS is constructed so its radius is the sum of the radii of the two given circles. This means that JL = FP.
We construct the tangent PJ from the point P to the circle OJS. This is done using the method described in Tangents through an external point.
The desired tangent FL is parallel to PJ and offset from it by JL. Since PJLF is a rectangle, we need the best way to construct this rectangle. The method used here is to construct PF parallel to OL using the "angle copy" method as shown in Constructing a parallel through a point

As shown below, there are two such tangents, the other one is constructed the same way but on the other half of the circles.

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

This is the same drawing as the last step in the above animation with line PJ added.

	Argument	Reason
1	PJ is a tangent to the outer circle O at J.	By construction. See Constructing the tangent through an external point for method and proof.
2	FP is parallel to LJ	By construction. See Constructing a parallel (angle copy method) for method and proof.
3	FP = LJ	QS was set from the radius of circle P in construction steps 2 and 3.
4	FPJL is a rectangle	FP is parallel to and equal to LJ from (2) and (3). ∠FLJ = ∠FLO = 90° (a tangent is at right angles to radius)
5	∠PFL = ∠FLO = 90°	Interior angles of rectangles are 90° (4)
6	FL is a tangent to circle O and P	Touches each circle at one place (F and L), and is at right angles to the radius at the point of contact, (5)

- Q.E.D

Click here for a printable tangents to two circles construction worksheet with some problems to try. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.