Let $S$ be a circle with center $O$. Suppose $A , B$ are points on the circumference of $S$ with $\angle A O B = 120 ^ { \circ }$. For triangle $A O B$, let $C$ be its circumcenter and $D$ its orthocenter (i.e., the point of intersection of the three lines containing the altitudes). For each statement below, write whether it is TRUE or FALSE. a) The triangle $A O C$ is equilateral. Answer: $\_\_\_\_$ b) The triangle $A B D$ is equilateral. Answer: $\_\_\_\_$ c) The point $C$ lies on the circle $S$. Answer: $\_\_\_\_$ d) The point $D$ lies on the circle $S$. Answer: $\_\_\_\_$
Let $S$ be a circle with center $O$. Suppose $A , B$ are points on the circumference of $S$ with $\angle A O B = 120 ^ { \circ }$. For triangle $A O B$, let $C$ be its circumcenter and $D$ its orthocenter (i.e., the point of intersection of the three lines containing the altitudes). For each statement below, write whether it is TRUE or FALSE.\\
a) The triangle $A O C$ is equilateral.
Answer: $\_\_\_\_$\\
b) The triangle $A B D$ is equilateral.
Answer: $\_\_\_\_$\\
c) The point $C$ lies on the circle $S$.
Answer: $\_\_\_\_$\\
d) The point $D$ lies on the circle $S$.
Answer: $\_\_\_\_$