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In this note, we show a sharpened version of the classical fundamental triangle inequality, as follows 2R2+10Rr−r2−2(R−2r) √ R2 − 2Rr cos φ s 2R2+10Rr−r2+2(R−2r) √ R2 − 2Rr cos φ, where φ = min 1 i<j 3 ∣ ∣Ai − Aj ∣ ∣ . Mathematics subject classification (2000): 26D05, 26D15, 51N35.

A sharpened version of the fundamental triangle inequality