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Question Number 39458 by MJS last updated on 06/Jul/18

$$\mathrm{find}\mathrm{the}\mathrm{greatest}\mathrm{possible}\mathrm{square}\mathrm{insribed}\mathrm{in}$$$$\mathrm{a}\mathrm{triangle}\mathrm{with}\mathrm{sides}abc$$

Commented by MJS last updated on 06/Jul/18

$$\mathrm{the}\mathrm{side}s\mathrm{of}\mathrm{the}\mathrm{square}\mathrm{not}\mathrm{necessarily}\mathrm{on}$$$$\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{sides}\mathrm{of}\mathrm{the}\mathrm{triangle}$$

Commented by ajfour last updated on 06/Jul/18

$$In△PCR$$$$\frac{s\sqrt{2}}{\sin \gamma }=\frac{b-y}{\sin (\theta +\frac{\pi }{4})}....\left(i\right)$$$$In△SRA$$$$\frac{s}{\sin \alpha }=\frac{y}{\sin (\beta -\theta )}.....\left(ii\right)$$$$eliminatingyamongst\left(i\right),\left(ii\right)$$$$\frac{s\sqrt{2}\sin (\theta +\frac{\pi }{4})}{\sin \gamma }=b-\frac{s\sin (\beta -\theta )}{\sin \alpha }$$$$\Rightarrow \mathit{s}=\frac{\mathit{b}}{\frac{\sqrt{2}\sin (\mathit{\theta }+\frac{\mathit{\pi }}{4})}{\sin \mathit{\gamma }}+\frac{\sin (\mathit{\beta }-\mathit{\theta })}{\sin \mathit{\alpha }}}$$$$=\frac{b}{\frac{\sin \theta +\cos \theta }{\sin \gamma }+\frac{\sin \beta \cos \theta -\cos \beta \sin \theta }{\sin \alpha }}$$$$=\frac{b\sin \gamma \sin \alpha }{\sin \theta (\sin \alpha -\cos \beta \sin \gamma )+\cos \theta (\sin \alpha +\sin \beta \sin \gamma )}$$$$\mathit{s}=\frac{b\sin \gamma \sin \alpha }{A\sin (\theta +\varphi )}$$$$\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}$$$$\tan \mathit{\varphi }=\frac{\sin \alpha +\sin \beta \sin \gamma }{\sin \alpha -\cos \beta \sin \gamma }$$$$\mathit{A}=\sqrt{{(\sin \alpha -\cos \beta \sin \gamma )}^2+{(\sin \alpha +\sin \beta \sin \gamma )}^2}$$$$forstobemaximum$$$$\sin (\theta +\varphi )hastobeminimum.$$$$Formanyofthecases(though$$$$notall),$$$$\sin (\theta +\varphi )isminimumif\theta =0$$$$Then$$$${\mathit{s}}_{max}=\frac{\mathit{b}\sin \mathit{\gamma }\sin \mathit{\alpha }}{A\sin \mathit{\varphi }}$$$$=\frac{b\sin \gamma \sin \alpha }{\sin \alpha +\sin \beta \sin \gamma }$$$$andsince$$$$\frac{\sin \alpha }{a}=\frac{\sin \beta }{b}=\frac{\sin \gamma }{c}=\frac{1}{2R}$$$${\mathit{s}}_{max}=\frac{abc}{2aR+bc}=\frac{a}{\frac{2aR}{bc}+1}.$$$$Thisformulawillhavetobe$$$$amendedif\sin (\theta +\varphi )<\sin \varphi$$$$fornonzero\theta .$$

Commented by ajfour last updated on 06/Jul/18

Commented by MJS last updated on 06/Jul/18

$$\mathrm{thanks}.\mathrm{I}\mathrm{already}\mathrm{thought}\mathrm{there}\mathrm{would}\mathrm{be}$$$$\mathrm{several}\mathrm{different}\mathrm{cases}...$$

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