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Question Number 25728 by ajfour last updated on 13/Dec/17

Commented by ajfour last updated on 13/Dec/17

$$Ifratioofareasof△PQRand$$$$△ABCbeindependentofa,b,c;$$$$findtheratio.$$

Commented by behi.8.3.4.17@gmail.com last updated on 14/Dec/17

Commented by mrW1 last updated on 14/Dec/17

$$verynice!$$

Answered by mrW1 last updated on 13/Dec/17

$$letBC=aandRQ=a^{\prime }etc.$$$$AR=AQ=\frac{b+c-a}{2}$$$$a^2=b^2+c^2-2bc\cos A$$$$a^{\prime 2}=2{\left(\frac{b+c-a}{2}\right)}^2[1-\cos A]$$$$=\frac{{(b+c-a)}^2}{2}[1-\frac{b^2+c^2-a^2}{2bc}]$$$$=\frac{{(b+c-a)}^2}{2}\left[\frac{a^2-{(b-c)}^2}{2bc}\right]$$$$=\frac{{(b+c-a)}^2(a+b-c)(a+c-b)}{4bc}$$$$\Rightarrow a^{\prime }=\frac{1}{2}\sqrt{\frac{{(b+c-a)}^2(a+b-c)(a+c-b)}{bc}}$$$$\Rightarrow a^{\prime }=2(s-a)\sqrt{\frac{(s-c)(s-b)}{bc}}$$$$\Rightarrow a^{\prime }=2(s-a)\sqrt{(\frac{s}{b}-1)(\frac{s}{c}-1)}$$$$\Rightarrow b^{\prime }=2(s-b)\sqrt{(\frac{s}{a}-1)(\frac{s}{c}-1)}$$$$\Rightarrow c^{\prime }=2(s-c)\sqrt{(\frac{s}{a}-1)(\frac{s}{b}-1)}$$$$withs=\frac{a+b+c}{2}and$$$$s^{\prime }=\frac{a^{\prime }+b^{\prime }+c^{\prime }}{2}$$$$\Rightarrow A_{PQR}=\sqrt{s^{\prime }(s^{\prime }-a)(s^{\prime }-b^{\prime })(s^{\prime }-c^{\prime })}$$

Commented by ajfour last updated on 14/Dec/17

$$pleaseexplainline2Sir..$$

Commented by mrW1 last updated on 14/Dec/17

$$AP=AQ=z$$$$BP=BR=x$$$$CR=CQ=y$$$$x+y=a$$$$y+z=b$$$$z+x=c$$$$\Rightarrow x+y+z=\frac{a+b+c}{2}$$$$\Rightarrow -x+(x+y+z)=b$$$$\Rightarrow x=(x+y+z)-b=\frac{a+b+c}{2}-b=\frac{a+c-b}{2}$$$$\Rightarrow y=(x+y+z)-c=\frac{a+b+c}{2}-c=\frac{a+b-c}{2}$$$$\Rightarrow z=(x+y+z)-a=\frac{a+b+c}{2}-a=\frac{b+c-a}{2}$$

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