Question

d is the mid point of side bc of an isosceles triangle abc with ab=ac. prove that the circle drawn with either of the equal side as a diameter passes through the point d

Answer 1

Your answer
ABC is an isosceles triangle
so ab=ac
d is the midpoint of BC
construct the circle and let side ab be its diameter
let another point g other than d pass through the circle(assumed)
construct AG
so angle AGB=90' (diameter property)
so AG is perpendicular to seg BC
therefor AG divides side BC(ABC is isosceles and isosceles triangle property)
so g is midpoint of seg BC
but d is midpoint of seg BC(given)
so d and g are one and the same
so point d passes through the circle drawn keeping ab as the diameter
similarly it can be proved for the other side also