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if point P is any point inside △ABC with side lenghts a,b and c. prove that; ((PA)/a) + ((PB)/b) + ((PC)/c) ≥ (√(3 ))

$$ \\ $$$$\:\:\:\:\:\:\boldsymbol{{if}}\:\boldsymbol{{point}}\:\boldsymbol{{P}}\:\:\boldsymbol{{is}}\:\boldsymbol{{any}}\:\boldsymbol{{point}}\:\boldsymbol{{inside}}\:\:\: \\ $$$$\:\:\:\:\:\:\:\bigtriangleup\boldsymbol{{ABC}}\:\:\boldsymbol{{with}}\:\boldsymbol{{side}}\:\boldsymbol{{lenghts}}\:\boldsymbol{{a}},\boldsymbol{{b}}\:\boldsymbol{{and}}\:\boldsymbol{{c}}.\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\boldsymbol{{prove}}\:\boldsymbol{{that}}; \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\boldsymbol{{PA}}}{\boldsymbol{{a}}}\:+\:\frac{\boldsymbol{{PB}}}{\boldsymbol{{b}}}\:+\:\frac{\boldsymbol{{PC}}}{\boldsymbol{{c}}}\:\geqslant\:\sqrt{\mathrm{3}\:} \\ $$$$ \\ $$

Question Number 187752    Answers: 0   Comments: 5

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Question Number 187691    Answers: 0   Comments: 0

Misalkan P sebarang titik didalam 𝚫ABC sehingga PD, PE , dan PF masing masing tegak lurus dengan sisi BC,CA dan AB Jika panjang sisi BC,CA dan AB masing masing dinotasikan dengan a,b dan c. bila x,y dan z adalah sebarang bilangan real yang memenuhi xy + yz +xz ≥ 0 Tunjukkan bahwa; (y + z) ((PA)/a) + (z+x)((PB)/b) + (x + y) ((PC)/c) ≥ 2(√(xy + yz +zx))

$$\: \\ $$$$\:\:\:\boldsymbol{{Misalkan}}\:\boldsymbol{{P}}\:\boldsymbol{{sebarang}}\:\boldsymbol{{titik}}\:\boldsymbol{{didalam}}\:\boldsymbol{\Delta{ABC}}\:\:\boldsymbol{{sehingga}}\:\boldsymbol{{PD}},\:\boldsymbol{{PE}}\:,\:\boldsymbol{{dan}}\:\boldsymbol{{PF}}\:\: \\ $$$$\:\:\:\boldsymbol{{masing}}\:\boldsymbol{{masing}}\:\boldsymbol{{tegak}}\:\boldsymbol{{lurus}}\:\:\boldsymbol{{dengan}}\:\boldsymbol{{sisi}}\:\boldsymbol{{BC}},\boldsymbol{{CA}}\:\boldsymbol{{dan}}\:\boldsymbol{{AB}} \\ $$$$\:\:\:\boldsymbol{{Jika}}\:\boldsymbol{{panjang}}\:\boldsymbol{{sisi}}\:\boldsymbol{{BC}},\boldsymbol{{CA}}\:\boldsymbol{{dan}}\:\boldsymbol{{AB}}\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\boldsymbol{{masing}}\:\boldsymbol{{masing}}\:\boldsymbol{{dinotasikan}}\:\boldsymbol{{dengan}}\:\boldsymbol{{a}},\boldsymbol{{b}}\:\boldsymbol{{dan}}\:\boldsymbol{{c}}. \\ $$$$\:\:\:\boldsymbol{{bila}}\:\boldsymbol{{x}},\boldsymbol{{y}}\:\boldsymbol{{dan}}\:\boldsymbol{{z}}\:\boldsymbol{{adalah}}\:\boldsymbol{{sebarang}}\:\boldsymbol{{bilangan}}\:\boldsymbol{{real}}\:\boldsymbol{{yang}}\:\boldsymbol{{memenuhi}}\:\boldsymbol{{xy}}\:+\:\boldsymbol{{yz}}\:+\boldsymbol{{xz}}\:\geqslant\:\mathrm{0}\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{Tunjukkan}}\:\boldsymbol{{bahwa}}; \\ $$$$\:\:\:\:\left(\boldsymbol{{y}}\:+\:\boldsymbol{{z}}\right)\:\frac{\boldsymbol{{PA}}}{\boldsymbol{{a}}}\:+\:\left(\boldsymbol{{z}}+\boldsymbol{{x}}\right)\frac{\boldsymbol{{PB}}}{\boldsymbol{{b}}}\:\:+\:\left(\boldsymbol{{x}}\:+\:\boldsymbol{{y}}\right)\:\frac{\boldsymbol{{PC}}}{\boldsymbol{{c}}}\:\:\geqslant\:\mathrm{2}\sqrt{\boldsymbol{{xy}}\:+\:\boldsymbol{{yz}}\:+\boldsymbol{{zx}}}\:\:\: \\ $$$$ \\ $$

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