10.373 In a triangle with sides 6, 10, and 12 cm, a circle is inscribed. A tangent is drawn to the circle such that it intersects the two larger sides. Find the perimeter of the cut-off triangle.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_21_f024bf2ff7725246f3bfg-230.jpg?height=293&width=327&top_left_y=1308&top_left_x=130)
Let $M N$ be the tangent to the circle inscribed in the given $\triangle A B C, A B=10, B C=12, A C=6$.
Then $B M+B N+M N=$ $=(10-A M)+(12-C N)+(A M+C N-6)=16$ (by theorem $2, M N=A M+C N-A C)$.

Answer: 16 cm.