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Question Number 30719 by mrW2 last updated on 24/Feb/18

Commented by mrW2 last updated on 25/Feb/18

$$Thisquestionisnotvisibleanymore.$$$$Thequestionis:$$$$Findthesidelengthsof\mathrm{\Delta }PQRin$$$$termsofa,b,c.$$

Commented by mrW2 last updated on 24/Feb/18

$$\frac{a^{\prime }}{a}=\frac{b^{\prime }}{b}=\frac{c^{\prime }}{c}=k=\frac{a^2+b^2+c^2}{8\sqrt{s(s-a)(s-b)(s-c)}}$$$$pleaseconfirm.$$

Commented by ajfour last updated on 25/Feb/18

$$ThanksSir,solutionplease..$$

Commented by beh.i83417@gmail.com last updated on 25/Feb/18

$$.$$

Answered by mrW2 last updated on 25/Feb/18

Commented by mrW2 last updated on 25/Feb/18

$$a^{\prime }=RP=RF+FG+GP$$$$RF=\frac{IF}{\sin B}=\frac{AF-AI}{\sin B}=\frac{\frac{c}{2}-AE\cos A}{\sin B}=\frac{c-b\cos A}{2\sin B}$$$$FG=BF\sin B=\frac{c\sin B}{2}$$$$GP=\frac{GD}{\tan C}=\frac{BD-BG}{\tan C}=\frac{\frac{a}{2}-BF\cos B}{\tan C}=\frac{a-c\cos B}{2\tan C}$$$$$$$$\Rightarrow a^{\prime }=\frac{c-b\cos A}{2\sin B}+\frac{c\sin B}{2}+\frac{a-c\cos B}{2\tan C}$$$$\Rightarrow a^{\prime }=\frac{(c-b\cos A)\sin C+c{\sin }^{2}B\sin C+(a-c\cos B)\sin B\cos C}{2\sin B\sin C}$$$$\Rightarrow a^{\prime }=\frac{c\sin C-b\cos A\sin C+c{\sin }^{2}B\sin C+a\sin B\cos C-c\cos B\sin B\cos C}{2\sin B\sin C}$$$$\Rightarrow a^{\prime }=\frac{c\sin C-b\cos A\sin C+c\sin B(\sin B\sin C-\cos B\cos C)+a\sin B\cos C}{2\sin B\sin C}$$$$\Rightarrow a^{\prime }=\frac{a\sin B\cos C-b\cos A\sin C+c(\sin C+\cos A\sin B)}{2\sin B\sin C}$$$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=\frac{1}{\lambda }=2R=\frac{abc}{2\sqrt{s(s-a)(a-b)(s-c)}}$$$$\sin A=a\lambda ,\sin B=b\lambda ,\sin C=c\lambda$$$$\Rightarrow a^{\prime }=\frac{ab\cos C-bc\cos A+c(c+b\cos A)}{2bc\lambda }$$$$\Rightarrow a^{\prime }=\frac{ab\cos C+c^2}{2bc\lambda }$$$$\Rightarrow a^{\prime }=\frac{\frac{a^2+b^2-c^2}{2}+c^2}{2bc\lambda }$$$$\Rightarrow a^{\prime }=\frac{a^2+b^2+c^2}{4bc\lambda }$$$$\Rightarrow a^{\prime }=\frac{a^2+b^2+c^2}{4bc\times \frac{2\sqrt{s(s-a)(s-b)(s-c)}}{abc}}$$$$\Rightarrow a^{\prime }=\frac{a^2+b^2+c^2}{\frac{8\sqrt{s(s-a)(s-b)(s-c)}}{a}}$$$$\Rightarrow a^{\prime }=a\times \frac{a^2+b^2+c^2}{8\sqrt{s(s-a)(s-b)(s-c)}}$$$$similarly:$$$$\Rightarrow b^{\prime }=b\times \frac{a^2+b^2+c^2}{8\sqrt{s(s-a)(s-b)(s-c)}}$$$$\Rightarrow c^{\prime }=c\times \frac{a^2+b^2+c^2}{8\sqrt{s(s-a)(s-b)(s-c)}}$$

Commented by ajfour last updated on 26/Feb/18

$$Thankyousir,greatsolution,$$$$andwasmuchawaited!$$

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