Napoleon's Theorem

'Napoleon's Theorem' printed from https://nrich.maths.org/

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Triangle $ABC$ has equilateral triangles drawn on its edges. Points $P$, $Q$ and $R$ are the centres of the equilateral triangles. You can change triangle $ABC$ below by dragging the vertices and observe what happens to triangle $PQR$.

What can you prove about the triangle $PQR$?

Created with GeoGebra

NOTES AND BACKGROUND

There are many ways of proving this result. One way you might like to try involves tessellation.

(1) Draw any triangle, with angles $A, B$ and $C$ say.

(2) Draw equilateral triangles $T_1, T_2$ and $T_3$ on the three sides of $\Delta ABC$.

(3) Fit copies of the original triangle and $T_1, T_2$ and $T_3$ into a tessellation pattern so that, at each vertex of the tessellation, the angles are $A, B$ and $C$ and three angles of $60^o$ making an angle sum of $360^o$.

(4) Napoleon's Theorem can be proved by simple geometry using a small part of this pattern without even assuming that this tessellation extends indefinitely in all directions, which is intuitively obvious but requires advanced mathematics to prove it.

Van Aubel's Theorem is related to Napoleon's Theorem. Van Aubel's Theorem states that if four squares are drawn on the edges of any quadrilateral then the lines joining the centres of the squares on opposite edges are equal in length and perpendicular.

For an animated proof of Van Aubel's Theorem see http://agutie.homestead.com/files/vanaubel.html