compass and straightedge construction of parallel line
Draw a circle with center (http://planetmath.org/Center8) and intersecting at two points, one of which is .
Draw a second circle with center and the same radius as . This circle also intersects at two points, one of which is .
Draw a third circle with center and radius . Let be the intersection point of (drawn below in red) with (drawn below in green) which lies on the same side of as does. The line (drawn below in blue) is the required parallel to .
The green circle shows that and are congruent.
The black circle shows that and are congruent.
The red circle shows that and are congruent.
Since 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 , the straightedge is totally superfluous, and the points and 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:
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|
|Date of creation||2013-03-22 17:11:18|
|Last modified on||2013-03-22 17:11:18|
|Last modified by||pahio (2872)|
|Synonym||construction of parallel|
|Synonym||construction of parallel line|