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Question Number 198828 by a.lgnaoui last updated on 24/Oct/23

Show that: Area(blue circle)=Area(Green circle)

$$\:\:\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{th}{at}}: \\ $$$$\:\boldsymbol{{Area}}\left(\boldsymbol{{blue}}\:\boldsymbol{{circle}}\right)=\boldsymbol{{Area}}\left(\boldsymbol{{Green}}\:\:\boldsymbol{{circle}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Commented by a.lgnaoui last updated on 24/Oct/23

Answered by mr W last updated on 25/Oct/23

Commented by mr W last updated on 25/Oct/23

Commented by mr W last updated on 25/Oct/23

AD=2R+p BD=R−p CD=R+p AB=(√2)R BC=R AC=(√(R^2 +(2R)^2 ))=(√5)R cos α=((((√2)R)^2 +(R−p)^2 −(2R+p)^2 )/(2(√2)R(R−p))) =−((R+6p)/(2(√2)(R−p)))=−((1+6λ)/(2(√2)(1−λ))) with λ=(p/R) cos β=((R^2 +(R−p)^2 −(R+p)^2 )/(2R(R−p))) =((R−4p)/(2(R−p)))=((1−4λ)/(2(1−λ))) α+β=π+(π/4) cos (α+β)=−((√2)/2) −((√2)/2)=−((1+6λ)/(2(√2)(1−λ)))×((1−4λ)/(2(1−λ)))−((√(8(1−λ)^2 −(1+6λ)^2 ))/(2(√2)(1−λ)))×((√(4(1−λ)^2 −(1−4λ)^2 ))/(2(1−λ))) −((√2)/2)=((24λ^2 −2λ−1)/( 4(√2)(1−λ)^2 ))−((√(3(7−28λ−28λ^2 )(1−4λ^2 )))/( 4(√2)(1−λ)^2 )) 28λ^2 −10λ+3=(√(3(7−28λ−28λ^2 )(1−4λ^2 ))) 4(λ−1)^2 (112λ^2 −3)=0 ⇒λ=1 ⇒rejected ⇒112λ^2 =3 ⇒λ=(√(3/(112)))=((√(21))/(28)) ⇒p=(((√(21))R)/(28)) ✓ in similar way we can get q=(((√(21))R)/(28)). p=q=(((√(21))R)/(28)) ⇒blue circle=green circle

$${AD}=\mathrm{2}{R}+{p} \\ $$$${BD}={R}−{p} \\ $$$${CD}={R}+{p} \\ $$$${AB}=\sqrt{\mathrm{2}}{R} \\ $$$${BC}={R} \\ $$$${AC}=\sqrt{{R}^{\mathrm{2}} +\left(\mathrm{2}{R}\right)^{\mathrm{2}} }=\sqrt{\mathrm{5}}{R} \\ $$$$\mathrm{cos}\:\alpha=\frac{\left(\sqrt{\mathrm{2}}{R}\right)^{\mathrm{2}} +\left({R}−{p}\right)^{\mathrm{2}} −\left(\mathrm{2}{R}+{p}\right)^{\mathrm{2}} }{\mathrm{2}\sqrt{\mathrm{2}}{R}\left({R}−{p}\right)} \\ $$$$\:\:\:\:=−\frac{{R}+\mathrm{6}{p}}{\mathrm{2}\sqrt{\mathrm{2}}\left({R}−{p}\right)}=−\frac{\mathrm{1}+\mathrm{6}\lambda}{\mathrm{2}\sqrt{\mathrm{2}}\left(\mathrm{1}−\lambda\right)} \\ $$$${with}\:\lambda=\frac{{p}}{{R}} \\ $$$$\mathrm{cos}\:\beta=\frac{{R}^{\mathrm{2}} +\left({R}−{p}\right)^{\mathrm{2}} −\left({R}+{p}\right)^{\mathrm{2}} }{\mathrm{2}{R}\left({R}−{p}\right)} \\ $$$$\:\:\:\:=\frac{{R}−\mathrm{4}{p}}{\mathrm{2}\left({R}−{p}\right)}=\frac{\mathrm{1}−\mathrm{4}\lambda}{\mathrm{2}\left(\mathrm{1}−\lambda\right)} \\ $$$$\alpha+\beta=\pi+\frac{\pi}{\mathrm{4}} \\ $$$$\mathrm{cos}\:\left(\alpha+\beta\right)=−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}=−\frac{\mathrm{1}+\mathrm{6}\lambda}{\mathrm{2}\sqrt{\mathrm{2}}\left(\mathrm{1}−\lambda\right)}×\frac{\mathrm{1}−\mathrm{4}\lambda}{\mathrm{2}\left(\mathrm{1}−\lambda\right)}−\frac{\sqrt{\mathrm{8}\left(\mathrm{1}−\lambda\right)^{\mathrm{2}} −\left(\mathrm{1}+\mathrm{6}\lambda\right)^{\mathrm{2}} }}{\mathrm{2}\sqrt{\mathrm{2}}\left(\mathrm{1}−\lambda\right)}×\frac{\sqrt{\mathrm{4}\left(\mathrm{1}−\lambda\right)^{\mathrm{2}} −\left(\mathrm{1}−\mathrm{4}\lambda\right)^{\mathrm{2}} }}{\mathrm{2}\left(\mathrm{1}−\lambda\right)} \\ $$$$−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}=\frac{\mathrm{24}\lambda^{\mathrm{2}} −\mathrm{2}\lambda−\mathrm{1}}{\:\mathrm{4}\sqrt{\mathrm{2}}\left(\mathrm{1}−\lambda\right)^{\mathrm{2}} }−\frac{\sqrt{\mathrm{3}\left(\mathrm{7}−\mathrm{28}\lambda−\mathrm{28}\lambda^{\mathrm{2}} \right)\left(\mathrm{1}−\mathrm{4}\lambda^{\mathrm{2}} \right)}}{\:\mathrm{4}\sqrt{\mathrm{2}}\left(\mathrm{1}−\lambda\right)^{\mathrm{2}} } \\ $$$$\mathrm{28}\lambda^{\mathrm{2}} −\mathrm{10}\lambda+\mathrm{3}=\sqrt{\mathrm{3}\left(\mathrm{7}−\mathrm{28}\lambda−\mathrm{28}\lambda^{\mathrm{2}} \right)\left(\mathrm{1}−\mathrm{4}\lambda^{\mathrm{2}} \right)} \\ $$$$\mathrm{4}\left(\lambda−\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{112}\lambda^{\mathrm{2}} −\mathrm{3}\right)=\mathrm{0} \\ $$$$\Rightarrow\lambda=\mathrm{1}\:\Rightarrow{rejected} \\ $$$$\Rightarrow\mathrm{112}\lambda^{\mathrm{2}} =\mathrm{3}\:\Rightarrow\lambda=\sqrt{\frac{\mathrm{3}}{\mathrm{112}}}=\frac{\sqrt{\mathrm{21}}}{\mathrm{28}} \\ $$$$\Rightarrow{p}=\frac{\sqrt{\mathrm{21}}{R}}{\mathrm{28}}\:\checkmark \\ $$$${in}\:{similar}\:{way}\:{we}\:{can}\:{get}\:{q}=\frac{\sqrt{\mathrm{21}}{R}}{\mathrm{28}}. \\ $$$${p}={q}=\frac{\sqrt{\mathrm{21}}{R}}{\mathrm{28}}\: \\ $$$$\Rightarrow{blue}\:{circle}={green}\:{circle} \\ $$

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