Question Number 39303 by behi83417@gmail.com last updated on 05/Jul/18
Commented by behi83417@gmail.com last updated on 05/Jul/18
$$\boldsymbol{{equaliteral}}\:\boldsymbol{{triangle}}\:\boldsymbol{{with}}\:\boldsymbol{{sides}}: \\ $$$$\boldsymbol{{BC}}=\mathrm{1},\boldsymbol{{rotates}}\:\boldsymbol{{about}}\::\boldsymbol{{B}},\boldsymbol{{at}}\:\boldsymbol{{angle}}:\mathrm{15}^{\bullet} \\ $$$$\left.\mathrm{1}\right)\boldsymbol{{find}}\:\boldsymbol{{common}}\:\boldsymbol{{area}}\:\boldsymbol{{between}} \\ $$$$\boldsymbol{{rest}}\:\boldsymbol{{and}}\:\boldsymbol{{rotated}}\:\boldsymbol{{case}}. \\ $$$$\left.\mathrm{2}\right)\boldsymbol{{yes}}\:\boldsymbol{{or}}\:\boldsymbol{{no}}! \\ $$$$\boldsymbol{{common}}\:\boldsymbol{{region}}\:\boldsymbol{{is}}\:\boldsymbol{{a}}\:\boldsymbol{{cyclic}}\:\boldsymbol{{shape}}. \\ $$
Answered by MrW3 last updated on 05/Jul/18
Commented by MrW3 last updated on 05/Jul/18
$${let}\:{a}={side}\:{length}\:=\mathrm{1} \\ $$$${let}\:\theta={rotation}\:{angle}=\mathrm{15}° \\ $$$$ \\ $$$${BD}=\frac{\sqrt{\mathrm{3}}{a}}{\mathrm{2}\:\mathrm{cos}\:\frac{\theta}{\mathrm{2}}} \\ $$$$\alpha=\mathrm{30}°−\frac{\theta}{\mathrm{2}} \\ $$$$\beta=\mathrm{60}°+\theta \\ $$$$\Rightarrow\alpha+\beta=\mathrm{90}°+\frac{\theta}{\mathrm{2}} \\ $$$$\frac{{EB}}{\mathrm{sin}\:\left(\mathrm{180}°−\alpha−\beta\right)}=\frac{{BD}}{\mathrm{sin}\:\beta} \\ $$$$\Rightarrow{EB}=\frac{\sqrt{\mathrm{3}}{a}\:\mathrm{sin}\:\left(\alpha+\beta\right)}{\mathrm{2}\:\mathrm{cos}\:\frac{\theta}{\mathrm{2}}\:\mathrm{sin}\:\beta}=\frac{\sqrt{\mathrm{3}}{a}}{\mathrm{2}\:\mathrm{sin}\:\left(\mathrm{60}°+\theta\right)} \\ $$$${A}_{\Delta{DBE}} =\frac{\mathrm{1}}{\mathrm{2}}×{DB}×{EB}×\mathrm{sin}\:\alpha \\ $$$${A}_{{common}} =\mathrm{2}{A}_{\Delta{DBE}} ={DB}×{EB}×\mathrm{sin}\:\alpha \\ $$$$=\frac{\sqrt{\mathrm{3}}{a}}{\mathrm{2}\:\mathrm{cos}\:\frac{\theta}{\mathrm{2}}}×\frac{\sqrt{\mathrm{3}}{a}}{\mathrm{2}\:\mathrm{sin}\:\left(\mathrm{60}°+\theta\right)}×\mathrm{sin}\:\left(\mathrm{30}°−\frac{\theta}{\mathrm{2}}\right) \\ $$$$=\frac{\mathrm{3}{a}^{\mathrm{2}} \:\mathrm{sin}\:\left(\mathrm{30}°−\frac{\theta}{\mathrm{2}}\right)}{\mathrm{4}\:\mathrm{cos}\:\frac{\theta}{\mathrm{2}}\:\mathrm{sin}\:\left(\mathrm{60}°+\theta\right)} \\ $$$$={A}_{\mathrm{0}} \frac{\sqrt{\mathrm{3}}\:\mathrm{sin}\:\left(\mathrm{30}°−\frac{\theta}{\mathrm{2}}\right)}{\mathrm{cos}\:\frac{\theta}{\mathrm{2}}\:\mathrm{sin}\:\left(\mathrm{60}°+\theta\right)} \\ $$$$ \\ $$$${the}\:{common}\:{area}\:{is}\:{not}\:{cyclic}\:{since} \\ $$$$\mathrm{2}\beta=\mathrm{120}°+\mathrm{2}\theta\:{is}\:{not}\:{always}\:{equal}\:{to}\:\mathrm{180}°. \\ $$