Ans: D
I must be true. Note that $\Delta ABC$ is an isosceles triangle and $AD\perp BC$. Therefore, $D$ is the mid-point of $BC$. Hence, $AD$ is the perpendicular bisector of $BC$. Therefore, the circumcentre of $\Delta ABC$ lies on $AD$.
I must be true. Note that $\Delta ABC$ is an isosceles triangle and $AD\perp BC$. Therefore, $D$ is the mid-point of $BC$. Hence, $AD$ is the perpendicular bisector of $BC$. Therefore, the circumcentre of $\Delta ABC$ lies on $AD$.
II must be true. Note that $AD$ is an altitude of $\Delta ABC$. Therefore, the orthocentre of $\Delta ABC$ lies on $AD$.
III must be true. Note that $AD$ is an median of $\Delta ABC$. Therefore, the centroid of $\Delta ABC$ lies on $AD$.
