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Misalkan-P-sebarang-titik-didalam-ABC-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-




Question Number 187691 by normans last updated on 21/Feb/23
      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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