In the following figure prove that: $AB+AC>PB+PC.$
It seems that the triangle inequality is useless unless we add something to picture...
2026-04-08 11:09:13.1775646553
In the following figure prove that: $AB+AC>PB+PC$
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Let $BP\cap AC=\{Q\}$. Hence, by triangle inequality we obtain: $$AB+AC=AB+AQ+QC>BQ+QC=BP+PQ+QC>BP+PC$$ and we are done!