Construction
As mentioned, Euclid created the rhombus, a parallelogram before parallel lines had even been proven. The supplies back then consisted of a compass and a straightedge. Follow along with this video to see how a rhombus could have been formed with these tools.
Working backwards, the rhombus can be used to construct a parallel line through a given point P. Get out geogebra or a writing utensil, compass, and straightedge to test it out.
- Pick any point along line l to be point A
- Draw an arc centered at A with radius length equal to segment AP that intersects l at point B
- Create a similar arc with radius equal to segment AP's length through P and B with the center being a point opposite from A, call it point Q.
- Connect points to form segments AP, PQ, AB, BQ. This is now rhombus APQB. Because a rhombus is made with parallel opposite sides, the line that segment PQ is on is parallel to line l.
How can one be certain they are working with a rhombus? Figures are not always drawn to scale. This can be done by proving its properties. That a given quadrilateral is a parallelogram with perpendicular bisectors. Watch the following video example problem to see how.