[ Sum of angles in a triangle. Theorem about the exterior angle. ] [ Criteria and properties of an isosceles triangle. ]

In triangle $A B C$, angle $C$ is three times angle $A$. A point $D$ is taken on side $A B$ such that $B D=B C$.

Find $C D$, if $A D=4$.
Let $\angle A=\alpha, \angle A C D=\varphi$, then $\angle C=3 \alpha, \angle B C D=\alpha+\varphi$. Therefore, $\varphi+(\alpha+\varphi)=3 \alpha$, that is, $\varphi=\alpha$. Thus, triangle $A D C$ is isosceles with base $A C$. Therefore, $C D=A D=4$.

![](https://cdn.mathpix.com/cropped/2024_05_06_7041536e9c9bacd5befeg-59.jpg?height=294&width=594&top_left_y=502&top_left_x=732)

Answer

$C D=4$.