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.


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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