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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:
This problem gives an opportunity to find the parameters that express the generality in the context.

For example, if the distance between the feet of perpendiculars dropped from $A$ and $B$ is taken as the unit length, then the length of the perpendiculars can be expressed in terms of that unit.

There may be other useful unit lengths to consider, but the important skill is to see that enlarging the diagram doesn't change the feature being explored, so a thoughtful choice of unit can help to make the problem more convenient to handle and access.