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Home / Drawing a Line Tangent to Two Circles

Drawing a Line Tangent to Two Circles

This page shows how to draw 1 of the two possible external tangents common to ii given circles with compass and straightedge or ruler. This construction assumes you are already familiar with Constructing the Perpendicular Bisector of a Line Segment.

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

The construction has three main steps:

  1. The circle OJS is constructed and so its radius is the difference between the radii of the ii given circles. This ways that JL = FP.
  2. We construct the tangent PJ from the point P to the circle OJS. This is washed using the method described in Tangents through an external betoken.
  3. The desired tangent FL is parallel to PJ and offset from it by JL. Since PJLF is a rectangle, we demand the best way to construct this rectangle. The method used here is to construct PF parallel to OL using the "angle re-create" method as shown in Constructing a parallel through a point

Every bit shown beneath, there are two such tangents, the other ane is constructed the same way merely on the bottom half of the circles.

Proof

This is the same drawing every bit the concluding step in the above blitheness with line PJ added.

Argument Reason
one PJ is a tangent to the inner circle O at J. By construction. See Amalgam the tangent through an external point for method and proof.
ii FP is parallel to LJ By construction. Encounter Constructing a parallel (bending copy method) for method and proof.
three FP = LJ QS was gear up from the radius of circle P in construction steps ii and three.
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)
v ∠FLJ = ∠LFP = 90° Interior angles of rectangles are 90° (4)
six FL is a tangent to circumvolve 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.Eastward.D

Printable pace-past-step instructions

The to a higher place animation is available as a printable step-by-step educational activity sheet, which tin be used for making handouts or when a computer is not bachelor.

Try it yourself

Click hither for a printable tangents to 2 circles construction worksheet with some problems to endeavour. When you lot get to the page, apply the browser print control to print as many equally 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 north line segments
  • Deviation of ii line segments
  • Perpendicular bisector of a line segment
  • Perpendicular at a bespeak on a line
  • Perpendicular from a line through a signal
  • Perpendicular from endpoint of a ray
  • Split a segment into n equal parts
  • Parallel line through a bespeak (angle copy)
  • Parallel line through a point (rhombus)
  • Parallel line through a bespeak (translation)

Angles

  • Bisecting an angle
  • Copy an angle
  • Construct a 30° angle
  • Construct a 45° angle
  • Construct a sixty° bending
  • Construct a 90° angle (right angle)
  • Sum of n angles
  • Divergence of two angles
  • Supplementary angle
  • Complementary angle
  • Amalgam  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-sixty-xc triangle, given the hypotenuse
  • Triangle, given 3 sides (sss)
  • Triangle, given one side and next angles (asa)
  • Triangle, given two angles and not-included side (aas)
  • Triangle, given two sides and included angle (sas)
  • Triangle medians
  • Triangle midsegment
  • Triangle altitude
  • Triangle distance (exterior case)

Right triangles

  • Right Triangle, given one leg and hypotenuse (HL)
  • Correct Triangle, given both legs (LL)
  • Right Triangle, given hypotenuse and one angle (HA)
  • Correct 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
  • Circumvolve 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 1 side
  • Hexagon inscribed in a given circle
  • Pentagon inscribed in a given circumvolve

Non-Euclidean constructions

  • Construct an ellipse with string and pins
  • Find the middle of a circumvolve with any right-angled object

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