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Question Number 59516 by naka3546 last updated on 11/May/19

Answered by tanmay last updated on 11/May/19

$$△=\frac{1}{2}r(a+b+c)=\sqrt{s(s-a)(s-b)(s-c)}$$$$s=\frac{a+b+c}{2}=\frac{2x+2y+2z}{2}=x+y+z$$$$$$$$\frac{x+y+z}{3}⩾{\left(xyz\right)}^{\frac{1}{3}}$$$$x+y+z⩾3{\left(xyz\right)}^{\frac{1}{3}}$$$$\frac{a+b+c}{2}⩾3{\left(xyz\right)}^{\frac{1}{3}}$$$$\frac{a+b+c}{3}⩾2{\left(xyz\right)}^{\frac{1}{3}}$$$$\frac{a+b+c}{3}⩾{\left(abc\right)}^{\frac{1}{3}}$$$${\left(abc\right)}^{\frac{1}{3}}⩾2{\left(xyz\right)}^{\frac{1}{3}}$$$$abc⩾8xyz$$$$...$$$$...$$$$$$$$$$$$$$$$$$$$$$

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