Constructing the external tangents to two given circles.

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We start with the given circles O and P.

Note: If you are not given the center of a circle, you can find it using the method shown in Finding the center of a circle with compass and straightedge.
In steps 1-4, we construct a circle, center O, whose radius is the difference of the radii of the two given circles.
1.  Draw a line through the center points of both circles.

2.  Set the compass width to the radius of smaller circle P.

3.   Move the compasses to left edge of circle O and mark the point S inside the circle where it crosses the line PO.

4.   With the compasses on O, set the compass width to S and draw a circle.

This circle has a radius which is the difference of the radii of circles O and P

In steps 5-7, we construct the tangent from P to the inner circle O.
For more on this see Tangents through an external point.
(Since it is not necessary, we do not actually draw this tangent. It would be the line from P to J).
5.  Construct the perpendicular bisector of OP to find the midpoint M.

For more on this see Constructing a perpendicular at a point on a line.

6.   Set the compass on M and set its width to O.
7.   Make an arc across the inner circle, creating the point of tangency J.

8.   Draw a line from O through J, creating point L on the outer, given circle O.

This will become the point of tangency for the desired tangent line.

We next construct a radius of circle P which is parallel to the line OL. This is done using the "angle copy" method of constructing a tangent through an external point.
9.   Set the compasses width to OS, and draw an arc from P.
10.   Set the compass width to SJ.
11.   Draw an arc from Z across the previous arc, creating point T.

12.   Draw a line from P through T, creating point F where it crosses the given circle P.

F will become the point of tangency for the desired tangent line.

13.   Draw a line through F and L

Done.   FL is one of the two tangents common to the given circles.

The other tangent is at the bottom of the circles and is constructed in a similar way.

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