Next we will see two possible proofs of Morley’s Theorem. The first is an indirect proof, since, instead of constructing the adjacent angle trisectors’ intersection points and checking that these are the vertices of an equilateral triangle, we first construct an equilateral triangle and then we check that this is in fact obtained by connecting the intersections point of the adjacent angle trisectors. This proof has the advantage of being purely geometrical and does not require any knowledge of trigonometry, unlike most direct proofs.

1st Proof

On the other hand, with the help of trigonometry (in particular, the law of sines), it can be shown that the sides of the triangle obtained by the intersection of adjacent angle trisectors, besides being equal, have length equal to \(8R\sin \frac{a}{3} \sin \frac{b}{3} \sin \frac{c}{3} \), where \(a\), \(b\) and \(c\) are the values of the angles of the initial triangle and \(R\) is its circumradius. Note that, although in this proof it is only shown that one side of Morley's triangle has the referred length, the same argument proves that the formula is valid for the other two sides (in fact, a permutation of the variables \(a\), \(b\) and \(c\) in the given formula doesn't change it). Obviously, this means that all sides of Morley’s triangle are equal (that is, the triangle is equilateral).

2nd Proof