10.372 The bisector of angle $A$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$. Find the length of the chord $D C$, if the center of the circle inscribed in the given triangle is at a distance $\boldsymbol{n}$ from point $D$.
Solution.

![](https://cdn.mathpix.com/cropped/2024_05_21_f024bf2ff7725246f3bfg-230.jpg?height=335&width=329&top_left_y=661&top_left_x=121)
D
Let $O$ be the center of the circle inscribed in the given $\triangle ABC$. $\angle BCD = \angle BAD$ (Theorem 6) $\Rightarrow$ $\Rightarrow \angle BCD + \angle OCB = \angle CAD + \angle ACO = \angle COD$ (external angle) $\Rightarrow \angle OCD = \angle COD \Rightarrow OD = CD = n$.

## Answer: $n$.