Pentagon

Find the vertices of a pentagon given the midpoints of its sides.

Triangle Mid Points

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Pareq Exists

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

Two Points Plus One Line

Stage: 4 Challenge Level:
Simple hint: how would you find the set of points that are equidistant from two given points?

When might that locus not intersect with the given line?

Harder, last part: try dropping perpendiculars from $A$ and $B$ onto the given line.

The distance between those positions might be useful.

When is $P$ between those positions and when is it outside?