Cho $\Delta ABC$. CMR $(p-a)(p-b)(p-c)\leq \frac{1}{8}abc$
$(p-a)(p-b)(p-c)\leq \frac{1}{8}abc$
Started By Issac Newton, 09-03-2013 - 08:22
#1
Posted 09-03-2013 - 08:22
#2
Posted 09-03-2013 - 09:19
Ta có $(p-a)(p-b)(p-c)\leq \frac{1}{8}abc$Cho $\Delta ABC$. CMR $(p-a)(p-b)(p-c)\leq \frac{1}{8}abc$
$\Rightarrow (\frac{a+b+c}{2}-a)(\frac{a+b+c}{2}-b)(\frac{a+b+c}{2}-c)\leq \frac{1}{8}abc$
$(b+c-a)(a+c-b)(a+b-c)\leq abc$
Áp dụng bđt AMGM:
$(b+c-a)(a+c-b)\leq \frac{(b+c-a+a+c-b)^{2}}{4}\leq c^{2}$
Thực hiện 2 bđt tương tự, rồi nhân 3 bđt trên theo vế, rồi khai căn bậc 2, ta được đpcm
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