compass and straightedge construction of parallel line

Task.  Construct the line parallelMathworldPlanetmathPlanetmath to a given line and passing through a given point P which is not on .



  1. 1.

    Draw a circle c1 with center ( P and intersecting at two points, one of which is A.

  2. 2.

    Draw a second circle c2 with center A and the same radius r as c1. This circle also intersects at two points, one of which is B.

  3. 3.

    Draw a third circle c3 with center B and radius r.  Let C be the intersection point of c3 (drawn below in red) with c1 (drawn below in green) which lies on the same side of as P does.  The line PC (drawn below in blue) is the required parallel to .


Note 1.  The construction is based on the fact that the quadrilateralMathworldPlanetmath PABC is a parallelogramMathworldPlanetmath. In fact, PABC is a rhombusMathworldPlanetmath. The reasoning is as follows:

  • The green circle shows that PC¯ and PA¯ are congruent.

  • The black circle shows that PA¯ and AB¯ are congruent.

  • The red circle shows that AB¯ and BC¯ are congruent.

  • Since  PABC  is a quadrilateral with all sides congruent, it is a rhombus (and therefore a parallelogram).

Note 2.  It is clear that the construction only needs the compass, not a straightedge: In determining the point C, the straightedge is totally superfluous, and the points P and C determine the desired line (which thus is not necessary to actually draw!). It may be proved that all constructions with compass and straightedge are possible using only the compass.

Note 3.  Another construction of the parallel uses the fact that the endpointsMathworldPlanetmath of two congruent chords (red) in a circle determine two parallel chords:


If you are interested in seeing the rules for compass and straightedge constructions, click on the provided.

Title compass and straightedge construction of parallel line
Canonical name CompassAndStraightedgeConstructionOfParallelLine
Date of creation 2013-03-22 17:11:18
Last modified on 2013-03-22 17:11:18
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 24
Author pahio (2872)
Entry type Algorithm
Classification msc 51-00
Classification msc 51M15
Synonym construction of parallel
Synonym construction of parallel line
Related topic ParallelPostulate
Related topic NSectionOfLineSegmentWithCompassAndStraightedge