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You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?
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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.
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.
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.
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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.