### Triangle Midpoints

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.

### The Medieval Octagon

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

# Two Points Plus One Line

##### Age 14 to 16 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?