Prove that area of DEF = area of D'E'F'

Question

Question -

If D,D' E,E' F,F' are isotomics pairs of points on side BC,CA,AB of ABC Prove that area of DEF = D'E'F'

My try -

Because they are isotomics they are equidistant from midpoint of corresponding sides..so I First try to prove that they are congruent but unable to do so ..any hints will be valuable..

Answer 1

Hint: Perform an affine transformation so that the triangle is equilateral.

Then the result follows since

$$ A( 1-B) + B ( 1 - C) + C ( 1-A) = A ( 1-C) + B (1-A)+ C ( 1- B) $$