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