Prove thatIf the bisector of any angle of a triangle and the perpendicular bisector ofits opposite side intersect, they will intersect on the circumcircle of thetriangle.

Prove thatIf the bisector of any angle of a triangle and the perpendic

1.	Prove thatIf the bisector of any angle of a triangle and the perpendicular bisector ofits opposite side intersect, they will intersect on the circumcircle of thetriangle.
Answer» Given `DeltaABC` is inscribed in a circle. Bissecter of `angleA` and perpendicular bisector of BC intersect at point Q. To prove A, B, Q and C are non-cyclic. Constrution Join BQ and QC. Proof We have assumed that, Q lies outside the circle. In `DeltaBMQ and DeltaCMQ`, BM=CM [QM is the perpendicular bisector of BC] `angleBMQ=angleCMQ ["each" 90^(@)]` MQ=MQ [common side] `:. DeltaBMQ cong DeltaCMQ` [by SAS congruence rule] `:. BQ=CQ` [by CPCT]..(i) Also, `angleBAQ=angleCAQ` [given]..(ii) From Eqs. (i) and (ii), we cen say that Q lies on the circle. [equal chords of a circle subtend equal angles at the circumference.] Hence, A, B, Q and C are non-cyclic.

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Prove thatIf the bisector of any angle of a triangle and the perpendicular bisector ofits opposite side intersect, they will intersect on the circumcircle of thetriangle.

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