Q. 75.0( 1 Vote )

In the picture, p

Answer :

Given: In the picture, points P, Q, R are marked on the sides BC, CA, AB of Δ ABC and the circumcircles, of Δ AQR, Δ BRP, Δ CPQ are drawn.


To Prove: All the circles pass through a common point.


Proof:



Let us assume a point O, which passes through all the circles.


and Join OP, OQ and OR.


If we prove that such a point O exists, then we are done.


Now, OQAR, OPCQ and OPBR are cyclic quadrilaterals, and in a cyclic quadrilateral, sum of any pair of opposite angles is 180°.


QOR + A = 180° …[1]


POQ + C = 180° …[2]


POR + B = 180° …[3]


Also, By angle sum property of triangle


A + B + C = 180° …[4]


Now, Adding [1] [2] and [3]


QOR + POQ + POR + A + C + B = 540°


POQ + QOR + POR + 180° = 540° [From 4]


POQ + QOR + POR = 360°


Now, sum of all the angles around O is 360°,


O exists and is common to all circles.


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