## Task A-1.4.

Let $D$ be the foot of the altitude from vertex $C$ of isosceles triangle $ABC$ with base $\overline{AB}$. Point $M$ is the midpoint of segment $\overline{CD}$. Lines $BM$ and $AC$ intersect at point $E$.

Determine the ratio $|CE|:|AC|$.
## Second Solution.

Let point $G$ be the midpoint of side $\overline{B C}$. 1 point

Let point $T$ be the intersection of segments $\overline{B M}$ and $\overline{D G}$.

![](https://cdn.mathpix.com/cropped/2024_05_30_9a2ee33c697c48e6349cg-07.jpg?height=631&width=888&top_left_y=433&top_left_x=515)

Since points $M$ and $G$ are the midpoints of sides $\overline{C D}$ and $\overline{B C}$, we conclude that point $T$ is the centroid of triangle $B C D$.

Therefore, $|G T|:|D G|=1: 3$.

Triangle $A B C$ is isosceles, so point $D$ is the midpoint of side $\overline{A B}$. Since point $G$ is also the midpoint of side $\overline{B C}$, we conclude that segment $\overline{D G}$ is the midline of triangle $A B C$.

Thus, lines $D G$ and $A C$ are parallel.

Finally, $|C E|:|A C|=|T G|:|D G|=1: 3$.