Morley's Trisector Theorem

Morley's Trisector Theorem states that the intersections of the adjacent pairs of angle trisectors of an arbitrary triangle are the vertices of an equilateral triangle.

The proof of this theorem involves the following steps.

1. Apply sine rule to \(\Delta ABR\) and compute \(AR\).

2. Simplify the expression for \(AR\) using the identity \(\sin 3\theta = 4\sin\theta\sin(60^{\circ}-theta)\sin(60^{\circ}+\theta)\)

3. Apply sine rule to \(\Delta ACQ\) and compute \(AQ\).

4. Simplify the expression.

5. Apply cosine rule to \(\Delta AQR\) to compute \(QR^{2}\).

6. Use the angle version of the cosine rule to the triangle with angles \(u\), \(v+60^{\circ}\) and \(w+60^{\circ}\) to simplify the expression for \(QR^{2}\). Note that \(u+60^{\circ}+v+60^{\circ}+w=180^{\circ}\).

7. As the expression is made symmetric about \(u\), \(v\) and \(w\), the triangle is equilateral.

There's a generalisation for this theorem, which involves about twenty seven different points formed by various intersections of internal as well as external trisectors.

Yeah.