MHB Proving isosceles using centroid and medians

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The discussion focuses on proving the isosceles triangle property using centroids and medians. By letting sides AB, AC, and BC be represented as c, b, and a respectively, the centroid properties are applied alongside Stewart's Theorem to derive an equation involving the triangle's sides. The resulting equation simplifies to $(b-c)^2 \left[ (b-c)^2-a^2 \right] =0$. The analysis reveals that the bracket fails the triangle inequality, leading to the conclusion that b must equal c, indicating that AB equals AC. The author expresses difficulty in finding a synthetic proof for this result.
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I can definitely do this in the opposite direction, but
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Letting $AB=c$, $AC=b$ and noting that the usual property of the centroid tells us that $GC = \frac{2}{3}$ of the median from $C$ and $GB = \frac{2}{3}$ of the median from $B$ we can use Stewart's Theorem to write everything in terms of the three sides $a,b,c$. It is an involved equation requiring two squarings to get rid of all radicals. I used Wolfram Alpha to double check my computations. After simplifying everything I end up with

$(b-c)^2 \left[ (b-c)^2-a^2 \right] =0$.

The bracket fails the triangle inequality which only leaves $b=c$ or $AB=AC$. I can't see any synthetic proof.
 
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