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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:

Do you know the formula for the area of a whole circle?

Use that to find the correct radius for this circular design.

There are three equal "tears", what angle of rotation do you need for one "tear" to match with its neighbour?

What angle do you have in an equilateral triangle?

How does that help with the angle you need to make at the centre?

Three semicircles arc from the edge to the centre of the design.

Where is the centre for each semi-circular arc, and how will you locate each point accurately?