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Let ABC be a triangle with $AB=c, BC=a, CA=b$. Let I be the incenter and G be the centroid of ABC. Assume that $GI$ perpendicular to $CI$. Prove that $6ab=(a+b)(a+b+c)$. I don't have any idea how to solve it.

The problem is the same as this one: How to prove that $ 6ab=(a+b)(a+b+c)$ for triangle

But I would like a synthetic proof without trilinear coordinates (meaning using only Euclid's axioms). I were triying many thing but I didn't prove it. I anyone has any idea please I don't want a full solution.

Thanks

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Weijie Chen
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  • A synthetic proof is where you start with known statements and build up towards your goal - but the answer here is synthetic. You should edit the old question, instead of repeating it again. – Dietrich Burde Jan 03 '16 at 19:55
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    @DietrichBurde With synthetic proof I mean only euclid axioms. I don't see how to write that without trilinear coordinates – Weijie Chen Jan 03 '16 at 19:58

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