11.3. The perimeter of triangle $ABC$ is 24 cm, and the segment connecting the point of intersection of its medians with the point of intersection of its angle bisectors is parallel to side $AC$. Find the length of $AC$.
Answer: 8 cm.

Solution. Let AK be the median from vertex A, M - the point of intersection of the medians ABC, and I - the point of intersection of its angle bisectors AA1, BB1, CC1. Draw a line through K parallel to AC, intersecting the angle bisector BB1 at point P - its midpoint. By Thales' theorem, $PI: IB1 = KM: MA = 2$, so $BI: IB1 = 1$. By the property of angle bisectors AI and CI in triangles $ABB1$ and $CBB1$, we have

![](https://cdn.mathpix.com/cropped/2024_05_06_c2c32f9c2afddeec3788g-2.jpg?height=548&width=754&top_left_y=754&top_left_x=1116)
$AB: AB1 = BI: IB1 = CB: CB1 = 1$. Therefore, $AC = \frac{1}{2}(AB + BC) = \frac{1}{3}(AB + BC + AC) = 8$.

Evaluation. Just the answer: 0 points. Finding the ratio $BI: IB1 = 1: 3$: 1 point. Finding the ratios $AB: AB1 = BI: IB1 = CB: CB1 = 1: 2$: 2 points.