On the sides of a triangle we construct equilateral triangles. The blue lines are of equal length and intersect in one point at an angle of 120 degrees.
For the proof, we write
APa = AB + d(BC),
where d is the rotation of 60 degrees. Thus
d(d(APa) = d(d(AB)) + d(d(d(BC) = d(BA) + CB = CPc.
We get the result.