How to Draw a Tanget Between Two Circles in Autocad

This page shows how to draw one of the two possible external 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 difference between 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 bottom half of the circles.

Proof

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 inner 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). ∠PJL = ∠PJO = 90° (a tangent is at right angles to radius)
5	∠FLJ = ∠LFP = 90°	Interior angles of rectangles are 90° (4)
6	FL is a tangent to circle O and P	Touches the circle at one place (F and L), and is at right angles to the radius at the point of contact

- Q.E.D

Printable step-by-step instructions

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.

Try it yourself

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.

Other constructions pages on this site

List of printable constructions worksheets

Lines

Introduction to constructions
Copy a line segment
Sum of n line segments
Difference of two line segments
Perpendicular bisector of a line segment
Perpendicular at a point on a line
Perpendicular from a line through a point
Perpendicular from endpoint of a ray
Divide a segment into n equal parts
Parallel line through a point (angle copy)
Parallel line through a point (rhombus)
Parallel line through a point (translation)

Angles

Bisecting an angle
Copy an angle
Construct a 30° angle
Construct a 45° angle
Construct a 60° angle
Construct a 90° angle (right angle)
Sum of n angles
Difference of two angles
Supplementary angle
Complementary angle
Constructing 75° 105° 120° 135° 150° angles and more

Triangles

Copy a triangle
Isosceles triangle, given base and side
Isosceles triangle, given base and altitude
Isosceles triangle, given leg and apex angle
Equilateral triangle
30-60-90 triangle, given the hypotenuse
Triangle, given 3 sides (sss)
Triangle, given one side and adjacent angles (asa)
Triangle, given two angles and non-included side (aas)
Triangle, given two sides and included angle (sas)
Triangle medians
Triangle midsegment
Triangle altitude
Triangle altitude (outside case)

Right triangles

Right Triangle, given one leg and hypotenuse (HL)
Right Triangle, given both legs (LL)
Right Triangle, given hypotenuse and one angle (HA)
Right Triangle, given one leg and one angle (LA)

Triangle Centers

Triangle incenter
Triangle circumcenter
Triangle orthocenter
Triangle centroid

Circles, Arcs and Ellipses

Finding the center of a circle
Circle given 3 points
Tangent at a point on the circle
Tangents through an external point
Tangents to two circles (external)
Tangents to two circles (internal)
Incircle of a triangle
Focus points of a given ellipse
Circumcircle of a triangle

Polygons

Square given one side
Square inscribed in a circle
Hexagon given one side
Hexagon inscribed in a given circle
Pentagon inscribed in a given circle

Non-Euclidean constructions

Construct an ellipse with string and pins
Find the center of a circle with any right-angled object

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Source: https://www.mathopenref.com/consttangentsext.html