Question

prove that the circle drawn on equal sides of an isosceles triangle as diameter bisect it

Answer 1

let the circle you draw side on AB of triangle ABC intersects the base BC at a point E and the center of the circle you draw be O. then
OB=OE=AB/2
so triangle OBE is isosceles and angle OBE = angle ABC
therefore triangles ABC and OBE are similar.(if two isosceles triangles have same base angle they are similar) 
then (OB/AB) =(BE/BC)
but (OB/AB)=1/2
therefore (BE/BC)=1/2
therefore E divides BC into two
HENCE PROVED.