Circle Tangents Questions (1) Triangle $ABC$, inscribed in a circle, has $AB = 15$ and $BC = 25$. A tangent to the circle is drawn at $B$, and a line through $A$ parallel to this tangent intersects $\overline{BC}$ at $D$. Find $DC$.

(2) Let the incircle of triangle $ABC$ be tangent to sides $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ at $D$, $E$, and $F$, respectively. Prove that triangle $DEF$ is acute.

I have tried several angle and arc formulas... but I can't get anywhere... 

HINTS.
(1) Triangles $CBA$ and $ABD$ are similar. Exploit that to find $BD$.
(2) The incenter of $ABC$ lies inside $DEF$, because it is the intersection of the angle bisectors.
