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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?

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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


Why do this problem 

This activity is plain straight forward problem solving. Here with traditional instruments. Working out what the task is asking for, doing something which might help but isn't the whole solution, reviewing the result of that to see what might bethere to be noticed, and then using that observation to close the task. Along the way working into a deeper appreciation of the things we thought we already knew and could do easily.

Possible approach 

This printable worksheet may be useful: Squirty.

Start the group off with the task of drawing a square of any size, using only a ruler and compasses. There's plenty to challenge understanding just in that.

Next have each student draw a triangle, pointing out that because they can each draw the triangle they choose, what follows will an exploration of any triangle.

Allow some discussion to establish that every student appreciates the task and that many 'first thoughts' about the task can be shared.

Pursue each suggestion, together as a group but with each students actually drawing for themselves. Exploring suggestions to their point of failure deepens understanding, skipping this reduces the value of the activity considerably.

If a prompt towards something fruitful is needed, point out that partially meeting criteria can sometimes be helpful in the problem-solving process. In this case to create a square, based on one side, one of whose 'top' corners touch one of the other two sides of the triangle. Several of these will be needed so that we can think about the special one of this set which touches both sides.

Take time to let students notice how they create their squares. How they choose the base or the 'touch point' and from that go on to create the rest of the square. Once they have three or four squares invite them to stand back, take a look and then share what they notice. Ask how that helps with the actual task

This is a relatively closed task but follow the extension suggestion to foster a classroom culture that 'plays' with compasses and sparks with geometric reasoning and insight.

Here's a couple of nice problems to start discussion and play.

  • If I only have the three mid-points from the sides of a triangle can I recreate the triangle ?

  • If I only have the centres of three circles where each circle touches the other two can I recreate the circles.

As before, these tasks are to use ruler and compasses, but do allow the discussion to wander into methods which depart from that constraint.

Also take time to allow students to appreciate what kind of constraint 'ruler and compasses' imposes.

'Ruler' means tha points can be joined and the line continued indefinitely from both ends, and 'compasses' means that a length can be picked up and transferred somewhere else.

Key questions 

  • Can you draw a square using only a straight edge and a pair of compasses ?
  • What, in your own words, is the challenge or task here ?


  • How might you start on that ? Can you do it 'straight off'' ?
  • How can you use that to get the square you want ?


Possible support 

Stick with creating squares just using ruler and compasses constructions.


Possible extension 

ZigZag explores more of what makes a square.