On Morley’s Trisector Theorem

Abstract

a þþ; b; c a; b þþ; c a; b; c þþ a; bþ; cþ aþ; b; cþ aþ; bþ; c þ; þ; þ because in each case these angles add to ða þ b þ cÞ þþ 1⁄4 180 . These triangles, so far determined only up to similarity, are illustrated in Figure 1. We can scale them so that the red lines in that figure all have the same length. The red lines from A to BC are the two lines through A that make angle aþ with BC , and are the same length since they form an isosceles triangle. (I call drawing such lines ‘‘dropping non-perpendiculars of angle aþ’’.) If one of the angles of ABC is obtuse—as is C in the figure—then the two angles at the feet of these non-perpendiculars (here cþ) are exterior angles of the isosceles triangle they form, rather than interior ones, but this does not affect the proof.