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.

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.

- 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 ?

What about midpoints of sides of other polyhedra?

Try the problem
Pentagon