Question 15
In plane, let be given a fixed circle $(O)$ with radius $R$, two fixed points $A$, $B$ on the circle $(O)$ such that $A$, $B$, $O$ are not collinear.
Let $C$ be a point on $(O)$ distinct from $A$ and $B$. Construct the circle $(O_1)$ passing through $A$, touching the line $BC$ at $C$, construct the circle $(O_2)$ passing through $B$, touching the line $AC$ at $C$. The circle $(O_1)$ and $(O_2)$ intersect again at $D$ distinct from $C$.
Prove that
$$
i) \, CD \le R;$$
$$ii) \text{The line } CD \text{passes through a fixed point when} C \text{moves on} (O) \text{ so that } C \text{ does not coincide with } A \text{ and } B.$$
($(O)$ denotes a circle with center $O$.)
Let $C$ be a point on $(O)$ distinct from $A$ and $B$. Construct the circle $(O_1)$ passing through $A$, touching the line $BC$ at $C$, construct the circle $(O_2)$ passing through $B$, touching the line $AC$ at $C$. The circle $(O_1)$ and $(O_2)$ intersect again at $D$ distinct from $C$.
Prove that
$$
i) \, CD \le R;$$
$$ii) \text{The line } CD \text{passes through a fixed point when} C \text{moves on} (O) \text{ so that } C \text{ does not coincide with } A \text{ and } B.$$
($(O)$ denotes a circle with center $O$.)