### 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.

### Folding Squares

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

# Triangle Midpoints

### Why do this problem ?

This problem has a number of possible routes to a solution. It could be used as the focus for applying or learning some standard 'ruler and compasses' techniques but it is also possible to answer the problem without them. One way to work at this is backwards, a good problem solving strategy! Try to leave plenty of room for different methods to emerge.

### Possible approach:

This printable worksheet may be useful: Triangle Midpoints.

To start with there is some thinking to do about the way in which knowledge of the mid points is sufficient to fix the triangle (surprising?). Show the question and ask for ideas about possible approaches, things the group notices and things that might be true but they/others would need to be convinced about.

One way forward might be to start from what the group knows or is relatively confident about. For example, being able to find the midpoints if you start with the triangle. It is this idea of working backwards, from what you know, that you are aiming to draw out of discussions.

Start by asking the group to imagine a triangle (in their mind's eye).
• What does it look like?
• Is it regular?
Encourage non-regular visualisations but this is not essential at this stage.
Now put a point at the midpoint of one side, then a second side and, finally, the third side.
Join the points up.
• What do you notice? The answer might just be they they make a triangle but press for any other ideas which might emerge that you could pursue.
You might wish to repeat this process. When ready, encourage groups to draw triangles of their own, to find midpoints and construct inner "midpoint triangles". After some time, share different results and anything the class has noticed, including a discussion of (and possibly practice at) how to find the midpoints. Learners may use paper folding or approximation so there is useful time to be spent on finding a midpoint accurately and discussing the alternatives.

Return to the problem, giving plenty of time for learners to find their own way. If additional stimulus is needed share ideas so far. You might offer the attached document to think about. To stimulate thinking about how you might convince someone that all four triangles are congruent this diagram might also be helpful.

Use time to discuss what different groups notice and how it might help. Encourage rigour.

This problem creates a good opportunity to practise some common classroom geometry constructions, for example: a perpendicular bisector of a given line, angle bisector centre for the circumcircle of a given triangle, drawing a line through a given point parallel to a given line.

It is particularly important to promote a lively discussion of possible reasons for a construction method's validity.

### Key questions:

• What does working backwards from what you know reveal about the problem that could be helpful?
• How do you know...? How can you convince us?
• Why does the construction work ?

### Possible extension:

What about midpoints of sides of other polyhedra?

### Possible support:

Establish the relationships with paper folding and cutting and usie this to see why triangles are congruent and therefore why particular approaches are valid.