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Question Number 39303 by behi83417@gmail.com last updated on 05/Jul/18

Commented by behi83417@gmail.com last updated on 05/Jul/18

$$\mathit{e}\mathit{q}\mathit{u}\mathit{a}\mathit{l}\mathit{i}\mathit{t}\mathit{e}\mathit{r}\mathit{a}\mathit{l}\mathit{t}\mathit{r}\mathit{i}\mathit{a}\mathit{n}\mathit{g}\mathit{l}\mathit{e}\mathit{w}\mathit{i}\mathit{t}\mathit{h}\mathit{s}\mathit{i}\mathit{d}\mathit{e}\mathit{s}:$$$$\mathit{B}\mathit{C}=1,\mathit{r}\mathit{o}\mathit{t}\mathit{a}\mathit{t}\mathit{e}\mathit{s}\mathit{a}\mathit{b}\mathit{o}\mathit{u}\mathit{t}:\mathit{B},\mathit{a}\mathit{t}\mathit{a}\mathit{n}\mathit{g}\mathit{l}\mathit{e}:{15}^{\bullet }$$$$1)\mathit{f}\mathit{i}\mathit{n}\mathit{d}\mathit{c}\mathit{o}\mathit{m}\mathit{m}\mathit{o}\mathit{n}\mathit{a}\mathit{r}\mathit{e}\mathit{a}\mathit{b}\mathit{e}\mathit{t}\mathit{w}\mathit{e}\mathit{e}\mathit{n}$$$$\mathit{r}\mathit{e}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{d}\mathit{r}\mathit{o}\mathit{t}\mathit{a}\mathit{t}\mathit{e}\mathit{d}\mathit{c}\mathit{a}\mathit{s}\mathit{e}.$$$$2)\mathit{y}\mathit{e}\mathit{s}\mathit{o}\mathit{r}\mathit{n}\mathit{o}!$$$$\mathit{c}\mathit{o}\mathit{m}\mathit{m}\mathit{o}\mathit{n}\mathit{r}\mathit{e}\mathit{g}\mathit{i}\mathit{o}\mathit{n}\mathit{i}\mathit{s}\mathit{a}\mathit{c}\mathit{y}\mathit{c}\mathit{l}\mathit{i}\mathit{c}\mathit{s}\mathit{h}\mathit{a}\mathit{p}\mathit{e}.$$

Answered by MrW3 last updated on 05/Jul/18

Commented by MrW3 last updated on 05/Jul/18

$$leta=sidelength=1$$$$let\theta =rotationangle=15°$$$$$$$$BD=\frac{\sqrt{3}a}{2\cos \frac{\theta }{2}}$$$$\alpha =30°-\frac{\theta }{2}$$$$\beta =60°+\theta$$$$\Rightarrow \alpha +\beta =90°+\frac{\theta }{2}$$$$\frac{EB}{\sin (180°-\alpha -\beta )}=\frac{BD}{\sin \beta }$$$$\Rightarrow EB=\frac{\sqrt{3}a\sin (\alpha +\beta )}{2\cos \frac{\theta }{2}\sin \beta }=\frac{\sqrt{3}a}{2\sin (60°+\theta )}$$$$A_{\mathrm{\Delta }DBE}=\frac{1}{2}\times DB\times EB\times \sin \alpha$$$$A_{common}=2A_{\mathrm{\Delta }DBE}=DB\times EB\times \sin \alpha$$$$=\frac{\sqrt{3}a}{2\cos \frac{\theta }{2}}\times \frac{\sqrt{3}a}{2\sin (60°+\theta )}\times \sin (30°-\frac{\theta }{2})$$$$=\frac{3a^2\sin (30°-\frac{\theta }{2})}{4\cos \frac{\theta }{2}\sin (60°+\theta )}$$$$=A_0\frac{\sqrt{3}\sin (30°-\frac{\theta }{2})}{\cos \frac{\theta }{2}\sin (60°+\theta )}$$$$$$$$thecommonareaisnotcyclicsince$$$$2\beta =120°+2\theta isnotalwaysequalto180°.$$

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