LT-MO 2006

Two circles are tangent externaly at a point $B$ . A line tangent to one of the circles at a point $A$ intersects the other circle at points $C$ and $D$ . Show that $A$ is equidistant to the lines $BC$ and $BD$ .

Find the maximal cardinality $|S|$ of the subset $S \subset A=\{1, 2, 3, \dots, 9\}$ given that no two sums $a+b | a, b \in S, a \neq b$ are equal.