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  • ABC is a given triangle. A line PQR cuts the sides of the triangle AB at P (when produced) AC at Q and BC at R. <img alt="\mathrm{PJ} \perp \mathrm{rBA}" src="https://entrancecorner.oncodecogs

ABC is a given triangle. A line PQR cuts the sides of the triangle AB at P (when produced) AC at Q and BC at R. \mathrm{PJ} \perp \mathrm{rBA} and \mathrm{\mathrm{QJ} \perp \mathrm{rAC}}. If line perpendicular upon BC through point R also passes from point R also passes from point J, find out the locus of the point J.

 

Option: 1

ellipse


Option: 2

Incircle \: of\: \triangle A B C


Option: 3

Circumcircle \: of \: \triangle A B C


Option: 4

pairs\: of\: \: straight\: lines


Answers (1)

best_answer

\angle \mathrm{PQJ}=\beta, \angle \mathrm{AQP}=\alpha, \angle \mathrm{RQC}=\alpha, \angle \mathrm{RQC}=\alpha, \alpha +\beta=90^{\circ}\angle \mathrm{RJC}=\alpha, hence \angle \mathrm{RCJ}=\beta

Hence \: \angle \mathrm{PAC}=\angle \mathrm{RCJ}

\Rightarrowexterior angle of a quadrilateral = interior opposite angle 

\Rightarrow ABCJ is cyclic

\Rightarrow J lies on the circum-circle of the \Delta\: ABC

\Rightarrow locus of J is the circum-circle of the triangle ABC.

Posted by

avinash.dongre

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