ILL 1984 Problem 32

Angles of a given triangle $ABC$ are all smaller than $120^\circ$. Equilateral triangles $AFB, BDC$ and $CEA$ are constructed in the exterior of $ABC$.

(a) Prove that the lines $AD, BE$, and $CF$ pass through one point $S.$

(b) Prove that $SD + SE + SF = 2(SA + SB + SC).$