(See the picture: angles o,x and . (this last symbol is a little dot!); Morley triangle M₁M₂M₃)
Since arc C₂B₂A₂ = arc B₁A₁C₁, we have o = x + .
In triangle ABC the angle at A is pi - 2x - . , the angle at B is 2. + x, the angle at C is x - .
With the horizontal lines, AB makes angle x + . , AC makes angle pi - x, BC makes angle pi - . ,
AM₁ makes angle (x + .) + (pi - 2x - .)/3 = pi/3 + x/3 + 2./3 , etc.
To prove that M₁M₃ is horizontal, we have to prove that the angle AM₁M₃ is equal to pi/3 + x/3 + 2./3 .
Now to calculate this angle, knowing the angles of triangle ABC, is an exercise in analytic geometry.