Let C be a variable interior point of a fixed segment AB. Equilateral triangles ACB′ and CBA′ are constructed on the same side and ABC′ on the other side of the line AB.
(a) Prove that the lines AA′,BB′ , and CC′ meet at some point P.
(b) Find the locus of P as C varies.
(c) Prove that the centers A′′,B′′,C′′ of the three triangles form an equilateral triangle.
(d) Prove that A′′,B′′,C′′ , and P lie on a circle. geometryEquilateral Triangle