Question 109
$\triangle ABC$ is inscribed in a circle $k$ with center $O$ so that $\angle ACB = 120^\circ$.
a. If $H$ is the orthocenter of $\triangle ABC$, prove that $A, B, O, H$ lie on a circle with center the midpoint of the arc $ACB$. (The orthocenter of $\triangle ABC$ is the intersection point of its three altitudes.)
b. If $G$ is the centroid of $\triangle ABC$, and $I$ is the incenter of $\triangle ABH$, prove that the points $O, G, I, H$ lie on a line. (The centroid of $\triangle ABC$ is the intersection point of its three medians: a median connects a vertex of $\triangle ABC$ with the midpoint of the opposite side; the incenter of $\triangle ABC$ is the intersection of its three angle bisectors.)
a. If $H$ is the orthocenter of $\triangle ABC$, prove that $A, B, O, H$ lie on a circle with center the midpoint of the arc $ACB$. (The orthocenter of $\triangle ABC$ is the intersection point of its three altitudes.)
b. If $G$ is the centroid of $\triangle ABC$, and $I$ is the incenter of $\triangle ABH$, prove that the points $O, G, I, H$ lie on a line. (The centroid of $\triangle ABC$ is the intersection point of its three medians: a median connects a vertex of $\triangle ABC$ with the midpoint of the opposite side; the incenter of $\triangle ABC$ is the intersection of its three angle bisectors.)