compass and straightedge construction
of parallel line∗
pahio†
2013-03-21 23:05:29
Task. Construct the line parallel to a given line ` and passing through a
given point P which is not on `.
P
`
.
Solution.
1. Draw a circle c1 with center P and intersecting ` at two points, one of
which is A.
.
P
.
A
`
∗ hCompassAndStraightedgeConstructionOfParallelLinei
.
created: h2013-03-21i
by:
hpahioi version: h39506i Privacy setting: h1i hAlgorithmi h51-00i h51M15i
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You can reuse this document or portions thereof only if you do so under terms that are
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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.
.
P
.
A
`
B
.
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 P C (drawn below in
blue) is the required parallel to `.
2
.
P
C
.
A
`
B
.
Note 1. The construction is based on the fact that the quadrilateral P ABC
is a parallelogram. In fact, P ABC is a rhombus. The reasoning is as follows:
• The green circle shows that P C and P A are congruent.
• The black circle shows that P A and AB are congruent.
• The red circle shows that AB and BC are congruent.
• Since P ABC 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 endpoints of two congruent chords (red) in a circle determine two parallel chords:
3
.
P
.
l
.
If you are interested in seeing the rules for compass and straightedge constructions, click on the link provided.
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