Angle Trisectors on Circumcircle

The problem below has been taken from a discussion on the geometry.research newsgroup.

While investigating the Morley triangle, I came upon this theorem which some might find interesting.

It has been suggested that this result might follow from Morleys theorem. So far, I have been unable to make this connection. Any thoughts?

Den Roussel

Discussion

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For example, we know that ∠BA'C' = C/3 + 60°. Denote the point of intersection of C₁A₂ and BB₁ as P. I am going to show that ∠BPC₁ = C/3 + 60°, from which C₁A₂||A'C', and similarly for other two sides.

∠BPC₁ equals the sum of angular measures of arcs BC₁ = 2·C/3 and A₂B₁ = (A + B)/3. Whence, ∠BPC₁ = 2C/3 + (A + B)/3 = C/3 + 60°. Q.E.D.

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