Recently I came across a super cute proof of Napoleon’s triangle theorem using complex numbers in Béla Bollobás’s book “The Art of Mathematics: Coffee time in Memphis”. Realizing the fact that I never knew it till now, and the fact that it is neither discussed in the standard textbooks on Euclidean geometry I read till day nor any online reading source (for instance Wikipedia doesn’t discuss this proof, however the Cut the Knot page on Napolean’s triangle discusses this proof), I think that despite being so beautiful this proof is not very popular (especially among math Olympians and other mathematics enthusiasts at the undergraduate level), and thus I decided to dedicate this blogpost for discussing the proof.
|Proposition 1 (Napoleon’s triangle theorem)|
Given any triangle , we construct equilateral triangles erected outwards (i.e., the points lie in the exterior of ) on the sides respectively. Let be the centroids of triangles respectively. Then must be an equilateral triangle.
Here is an illustration of how the setup of Napoleon’s triangle theorem looks like.
Here are equilateral triangles erected outwards on the sides of $\Delta ABC,$ and $A”, B”, C”$ are the centroids of $\Delta A’BC, B’CA, C’AB$ respectively; we wish to show that the dotted triangle (i.e., the triangle with its edges marked by dotted lines
Proof: We notice that a triangle is equilateral if and only if the angle between the sides and is and in other words is equilateral if and only if rotating the side by an angle of gives the side of this triangle.
So, it suffices to show that rotation of side of triangle gives the side of this triangle; in terms of complex numbers a rotation by an angle of basically corresponds to multiplication by the complex number
The idea is to write each side of triangle in terms of complex numbers and then verify the fact that multiplication by rotates the side to yield
Let and Then one has and Also one obtains and Thus we immediately get
whence (since and the product of two non-zero complex numbers must necessarily be non-zero).
Now, notice that
Hence, thus proving the claim.
EDIT: (Last updated on 7 June, 2021)
|Remark: Some sources believe that the proof discussed above was known much before Béla Bollobás’s discovery of the proof.|