Formal Proof of Morley’s Theorem The following four steps will guide you in proving Morley...

Formal Proof of Morley’s Theorem The following four steps will guide you in proving Morley’s Theorem referred to in Sections 1.1 and 1.4. Construct points

X, Q', R' as shown in the figure, where m∠XPQ' = m∠XPR' =30 ∆PQ'R' is equilateral. (Note that P is the incenter of ∆BCX and bisects ∠BXC.) Let Y and Z be the intersections of the altitudes of ∆PQ'R' from Q' and R' with rays and . From the angle measures to be obtained, m∠YR'Q' + m∠ZQ'R' > 180, so rays will meet at some point A'. Now complete the details in the following steps:

(1) m∠PQ'Z = 60 – C/3 and m∠PR'Y = 60 – B/3.

(Hint: m∠PQ'Z = m∠Q'PZ = 180 − m∠Q'PB = 180 – (m∠XPB + 30); m∠XPB = 180 – B/3 – m∠BXC = 90 + C/3.)

(2) m∠A' = A/3.

(3) m∠A'ZB = 60 + 2C/3 and m∠A'YC = 60 + 2B/3.

(Hint: m∠A'ZB = 2 ∙ m∠R'ZQ' = 2 ∙ (90 − m∠PQ'Z.)

(4) m∠A'BC = B andm∠A'CB = C, hence A' = A, Q' = Q, R' = R, and ∆PQR = ∆PQ'R' is equilateral. (To show m∠A'BC = B, construct ray that ray and m∠KBC = B; show that ray meets at some point W, and that m∠R'WZ = A/3 = m∠R'A'Z. :. W = A'. Similarly, m∠A'CB = C.)