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

Stage: 4 Challenge Level:

Using only compasses and a straight edge, construct this design so that each "tear" is $100 \; \text{cm}^2$.

So that your diagram is the right size you will need to calculate one length, to get you started.

Do this as a single calculation, then take that result as the initial setting for your compasses.

You can then create the design without further calculation or use of a ruler

(a protractor is not necessary for any part of the process).