Construction: Parallel Line

How to construct a line that is parallel to the given line: by using a compass and a straightedge (ruler).

Example

Given

Let's see how to construct a line
that is parallel to the given line
and that passes through
the given point not on the line.

Solution

Draw a line
that passes through
the given line and the given point.

Place the point of the compass
at the intersection of the lines.
(blue point)

Draw an arc
that passes through the lines.

Without changing the radius of the compass,
place the point of the compass
at the given point.

Draw an arc (blue arc)
just like the previous arc.

Place the point of the compass
at the intersection
of the gray line and the previous arc.
(blue point)

The radius is the distance
from the blue point
to the intersecting point (blue arrow)
of the given line and the blue arc.
(blue dashed arc)

Without changing the radius of the compass,
place the point of the compass
at the intersection
of the gray line and the upper arc.
(blue point)

Draw an arc
that passes through the upper arc.
(blue arc)

Draw a line
that passes through
the given point and the intersection of the arcs.

Then the blue line is parallel to the given line.