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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.

Three Tears

Age 14 to 16
Challenge Level
This is initially an exercise in "de-construction" to enable construction.

The first calculation may involve a little algebra to rearrange the familiar area formula or substitution and solving the equation.
Either way it is necessary to establish the radius which will give the correct area.

Marking off the six arcs around a circle circumference is an important formative experience.

The student needs to check that they understand how this "works", how the process amounts to six equilateral triangles.