Question

In any triangle ABC, if the angle bisector of `/_A`and perpendicular bisector of BCintersect, prove that they intersect on the circumcircle of the triangle ABC

Answer 1

Angle bisector`angleA`meets the circumcircle of triangleABC at point P
from diagram
`angleBAD=angleCAD`
so, ABD is a Cyclic Quadrilateral
`angleDBC=angleDAC=1/2angleA`
`angleDCB=angle BAD=1/2angleA`
`angleDBC=angleDCB`
`In triangleDCB`
BP and CD are equal
D lies on perpendicular bisector of BC