In the following figure, $\angle C A B=2 \cdot \angle C B A, A D$ is an altitude and $M$ is the midpoint of $A B$. If $A C=2 \text{ cm}$, find the length of the segment $D M$.

![](https://cdn.mathpix.com/cropped/2024_05_01_12f29789e10e6749ebb9g-27.jpg?height=365&width=700&top_left_y=510&top_left_x=755)

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Solution

Let $K$ be the midpoint of $A C$. Since the circle with center $K$ and diameter $A C$ passes through $D$, it follows that $C K=A K=D K=1 \mathrm{~cm}$. From this last equality, it follows that triangle $A D K$ is isosceles and thus $\angle K D A=\angle K A D$. The segment $K M$ is the midline of triangle $A B C$, implying that $K M \| B C$ and thus $\angle K M D=\angle C B A$. By the Exterior Angle Theorem,

$$
\angle D K M=\angle A D K-\angle D M K=\angle D M K
$$

Finally, since the last equalities show that triangle $D K M$ is isosceles, it follows that $D M=D K=1 \mathrm{~cm}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_12f29789e10e6749ebb9g-27.jpg?height=365&width=698&top_left_y=1599&top_left_x=759)