An appealing math problem, due to Ken Sloan:


Given three different parallel lines in the plane, construct an equilateral triangle with one vertex on each of the three lines.


Solution: see the picture below:




An analogous problem: Given three intersecting lines, no two the same, construct a non-degenerate equilateral triangle with one vertex on each line.


Solution: Again, by pictures. You first pick any point P on the thin purple line, and create a copy of the blue line, rotated 120 degrees. In this case, the rotated version is drawn with medium thickness. Let Q be the intersection of the three thin lines. Now look at the intersection of the thin red and medium blue lines; call it S. Draw a thick magenta line (rotated -120 from the thin magenta line) through S, and a thick red line (rotated -120 from the thin red line) through P. These meet at a vertex Q''. Draw the thick blue line through that vertex.


The points where the tick magenta meets thin blue, thick red meets thing magenta, and thick blue meets thin red --- these are the vertices of an equilateral triangle.


Reason: construct the other two medium-thick lines as shown in the diagram, and the three-fold symmetry is evident.