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Can you draw a square ABCD with BC on the base PR of the triangle and the vertex D on the side PQ using ruler and compasses?
What happens to the point A as you enlarge the square? The interactivity may help. Move Q to specify your triangle, and then move D to enlarge the square.
How can you use this to help you draw the square you require?
You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.