At the end of 19th century, the British mathematician Frank Morley discovered the following surprising property :

**The three points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle.**

There are many proofs of this result, but one of them given by Alain Connes is especially beautiful and interesting as a group theoretic property of the action of the affine group on the line. We propose an overview of his proof.

**AFFINE GROUP**

Let be a commutative field and the affine group over that is to say the group of mappings where and . In fact, can be represented by the group of 2×2 invertible matrices of the form

working as follows

Next, we define a group homomorphism where is the multiplicative group of nonzero elements of *k*

Clearly, is the group of translations of *k*. We denote the translational part by

Finally, when there is a unique fixed point or that we denote by

**CONNES’S THEOREM**

The proof of the Morley’s theorem will appear as a direct consequence of the following theorem. Although it could seem rather abstract at first look, we will see later that it is really the solution of the problem.

**Connes’s Theorem :** Let be such that , , and are not translations and let . The two following conditions are equivalent

(1)

(2) and where , ,

To prove this equivalence, we notice that

The outline of the proof can be summarized as follows

With the help of a computer algebra system (like the open source software Maxima) we compute with no effort

with

where are the “a and b” coefficients of , , respectively.

First, we prove that is the same as . For this, we simply check that (the equality before the last uses the homomorphism property of )

Second, we prove that is the same as . For this, we notice that . We can simplify *B* as follows

In his paper, Connes proposes the following factorization

The factor cannot be zero because and is not a translation (hypothesis of the theorem) which means . The same occurs for the factors and . Therefore, is equivalent to .

**MORLEY’S THEOREM**

We take and define , , as follows

- is the rotation of center and angle where
- is the rotation of center and angle where
- is the rotation of center and angle where

Now, consider the point of intersection of the trisectors of angles and closest to the side . The rotation transforms to and the rotation transforms to . Therefore is a fixed point of . Similarly, is a fixed point of and is a fixed point of .

Next, is a rotation with center and angle . It is equivalent to a product of two reflections . Similarly, is equivalent to a product of two reflections and is equivalent to a product of two reflections .

We deduce

Thus, we are in the conditions of the Connes’s theorem, which implies and . Since , the equality means that is the cube root of unity . The relation can be interpreted in the complex plane as follows

This means that vector is obtained by rotation of angle from vector . The same angle occurs in the two other cases which proves that the triangle is equilateral.

**REFERENCES**

- A New Proof of Morley’s Theorem, Alain Connes, IHES, 1998, pp. 43-46
- Newsletter European Mathematical Society, December 2004, Issue 54