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Napoleon's Theorem

Age 14 to 18
Challenge Level

Why do this problem?

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.
The problem Hexi-metry involves the Cosine Rule.
If you want to use complex numbers then try the problem Complex Rotations.

Possible extension

Try Pythagoras for Tetrahedron