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Question 10
In any triangle ABC, if the angle bisector of A and perpendicular bisector of BC intersect, prove that they intersect on the circum circle of the triangle ABC.


Solution


Let angle bisector of A intersect circumcircle of ΔABC at D.
Join DC and DB.
BCD=BAD
[Angles in the same segment]
BCD=BAD=12A
[AD is bisector of A]          ...(i)
Similarly, DBC=DAC=12A          ...(ii)
From (i) and (ii), DBC=BCD
BD=DC [sides opposite to equal angles are equal]
D lies on the perpendicular bisector of BC.
Hence, angle bisector of A and the perpendicular bisector of BC interset on the circumcircle of ΔABC.

Mathematics
DUMMY
Standard IX

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