Question

In any triangle ABC, if the angle bisector of angle A and perpendicular bisector of BC intersect,prove that they intersect on the circumference of the triangle ABC

Answer 1

Heya User,

--> We use Happy Happy Geometry ^_^ to prove it..

--> Take ΔABC, draw the circumcircle..
--> Let the perpendicular bisector of BC intersect the circumcircle on E and the angle bisector of A intersect the circumcircle at E' ...

--> Now, in ΔBEC OE is perpendicular bisector
=> ΔBEC is isosceles => ∠CBE = ∠BCE 

==> ∠EAB = ∠EAC --> { Angle in the same segment applied twice }

However, ∠E'AC = ∠E'AB { due to our assumption }
=> E' and E are same points  ^_^

Hence, we say that the perpendicular bisector and the angle bisector intersect on the circumcircle ... =_= w.r.t your question ....