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Question Number 176555 by Shrinava last updated on 21/Sep/22

$$\mathrm{ABCD}-\mathrm{convex}\mathrm{quadrilateral}$$$$\mathrm{M}\in \mathrm{Int}\left(\mathrm{ABCD}\right),\mathrm{F}-\mathrm{area},\mathrm{s}-\mathrm{semiperimetr}$$$$\mathrm{a},\mathrm{b},\mathrm{c}-\mathrm{sides}.\mathrm{Prove}\mathrm{that}:$$$$\frac{{\mathrm{MA}}^4}{\mathrm{b}}+\frac{{\mathrm{MB}}^4}{{\mathrm{c}}^4}+\frac{{\mathrm{MC}}^4}{\mathrm{d}}+\frac{{\mathrm{MD}}^4}{\mathrm{a}}⩾\frac{{2\mathrm{F}}^2}{\mathrm{s}}$$

Commented by mr W last updated on 21/Sep/22

$$a=AB,b=BCetc.?$$

Commented by Shrinava last updated on 21/Sep/22

$$\mathrm{yes}\mathrm{dear}\mathrm{professor}$$

Answered by mr W last updated on 24/Sep/22

Commented by Shrinava last updated on 27/Sep/22

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{professor}$$$$\mathrm{so}{\mathrm{that}}^{\prime }\mathrm{s}\mathrm{it}?$$

Commented by mr W last updated on 04/Oct/22

$$s=\frac{a+b+c+d}{2}$$$$F=\frac{MA\times MB\times \sin \alpha }{2}+\frac{MB\times MC\times \sin \beta }{2}+\frac{MC\times MD\times \sin \gamma }{2}+\frac{MD\times MA\times \sin \delta }{2}$$$$⩽\frac{1}{2}(MA\times MB+MB\times MC+MC\times MD+MD\times MA)$$$$⩽\frac{1}{2}(\frac{{MA}^2+{MB}^2}{2}+\frac{{MB}^2+{MC}^2}{2}+\frac{{MC}^2+{MD}^2}{2}+\frac{{MD}^2+{MA}^2}{2})$$$$=\frac{{MA}^2+{MB}^2+{MC}^2+{MD}^2}{2}$$$$\frac{2F^2}{s}⩽\frac{{({MA}^2+{MB}^2+{MC}^2+{MD}^2)}^2}{a+b+c+d}$$$$⩽\frac{{MA}^4}{b}+\frac{{MB}^4}{c}+\frac{{MC}^4}{d}+\frac{{MD}^4}{a}\ast )$$$$proved!✓$$$$$$$$\ast )seeChaucy-SchwarzInequality$$

Commented by Shrinava last updated on 24/Sep/22

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{professor}\mathrm{please}$$

Commented by mr W last updated on 04/Oct/22

$$yes.nowtheproofiscomplete.$$

Commented by mr W last updated on 04/Oct/22

Commented by Shrinava last updated on 08/Oct/22

$$\mathrm{perfect}\mathrm{dear}\mathrm{professor}\mathrm{thank}\mathrm{you}$$

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