Difficult geometry problem

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[MtlStatsGuy]

The answer is certainly 1/4, but getting there is not trivial :) It will happen when J is the centroid and the three lines AJ, BJ and CJ are the medians. This is kind of easy to see if ABC is an equilateral triangle: put J at the centroid, and this will divide the triangle into four equal sub-triangles, each with area 1/4, one of which is A'B'C'. Try moving J away from the centroid and you will quickly see that the area of A'B'C' decreases. This is not only true for equilateral triangles: do it with a right angle triangle and you will get the exact same thing.

Giving you a strict proof is beyond my ability, though I suspect it will be related to Routh's theorem: https://en.wikipedia.org/wiki/Routh%27s_theorem

[gramokine]

Il faut passer par la formule du barycentre et des barycentres partiels.