8,9

Given a triangle with sides 12, 15, 18. A circle is drawn that touches both smaller sides and has its center on the larger side. Find the segments into which the center of the circle divides the larger side of the triangle.
The center of the circle inscribed in an angle lies on the bisector of this angle.

## Solution

The center of the circle lies on the bisector of the larger angle. Therefore, the ratio of the required segments is equal to the ratio of the lateral sides, i.e., $\frac{4}{5}$. Consequently, the required segments are

$$
\frac{4}{9} \cdot 18=8, \frac{5}{9} \cdot 18=10
$$

## Answer

8 and 10.