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Challenge 378: Triple Triangle Teaser

Triangle ABC is isosceles with angle BAC = 20o and AB = AC.

Point D, which is closer to C than B, is positioned so that triangle ADC is also isosceles, with DA = DC.

Points B, C and F lie on the same straight line, with C in between B and F, and triangle CDF is equilateral.

Prove that AB = BC + DC.

Justify each step in your solution clearly!