Students can experiment with the interactivty, observe what remains
invariant as the inner triangle changes, make a conjecture and then
try to prove it.

There are several diffferent ways to prove this result.

One way uses only the Cosine Rule and the area formula for a
triangle. It is quite short as it produces a formula that is
entirely symmetric in $a, b$ and $c$, the lengths of the sides of
the inner triangle, and then uses a symmetry argument to complete
the proof. This in itself is a good thing for students to see and
be aware of.

An alternative method uses a tessellation with copies of $\Delta
ABC$ and three triangles drawn on the sides of $\Delta ABC$. This
uses only elementary geometry. There is a second interactivity to
aid students in visualising the tessellation and proving the result
by this method.

Either of these two methods provide a Stage 4 challenge.

Alternatively you can use vectors or complex numbers (a Stage 5
challenge).

### Possible approach

After the students have experimented with the interactivity
and made their conjectures, then the teacher can either let them
find their own ways of proving the result, or alternatively suggest
one of the methods according to what the students know and where
practice and further familiarity with a concept might be
useful.

### Key questions

What do you know about the centroid of an equilateral
triangle?

Can you find the distance from the vertex to the centroid of
an equilateral triangle.

Can you write the lengths of the segment joining two centroids
in terms of the side lengths and angles of the inner
triangle?

Can you use a symmetry argument?

### Possible support

Learners can use

GeoGebra to draw and investigate
their own dynamic diagram for this theorem. It is free software and
easy to use.

### Possible extension

Try

Thebault's Theorem or

Pythagoras for Tetrahedron