6. a) Points $A$ and $B$ move uniformly and with equal angular velocities along circles centered at $O_{1}$ and $O_{2}$, respectively (clockwise). Prove that vertex $C$ of the equilateral triangle $A B C$ also moves uniformly along some circle.

b) The distances from a fixed point $P$ on the plane to two vertices $A, B$ of an equilateral triangle $A B C$ are $A P=2$ and $B P=3$. Determine the maximum value that the distance $C P$ can have.
6. a) Let $\overrightarrow{\mathrm{O}_{1} \mathrm{O}_{3}}$ be the vector obtained by rotating $\overrightarrow{O_{1} O_{2}}$ by $60^{\circ}$ (in the same direction as the rotation that transforms $\overrightarrow{A B}$ into $\overrightarrow{A C}$). Points $A^{\prime}$ and $B^{\prime}$ are the images of $A$ and $B$ under the same rotation around $O_{1}$. When vectors $\overrightarrow{O_{1} A}$ and $\overrightarrow{O_{2} B}$ rotate uniformly with the same angular velocity, $\triangle O_{1} A^{\prime} A$ will rotate around $O_{1}$, vectors $\overrightarrow{O_{3} B^{\prime}}$ and $\overrightarrow{B^{\prime} C}=\overrightarrow{A^{\prime} A}$, and thus their sum $\overrightarrow{O_{3} C}$ will rotate around $O_{8}$ (Fig. 23).

![](https://cdn.mathpix.com/cropped/2024_05_21_7b8a6b2625543c3dae75g-108.jpg?height=492&width=446&top_left_y=591&top_left_x=214)

Fig. 23

![](https://cdn.mathpix.com/cropped/2024_05_21_7b8a6b2625543c3dae75g-108.jpg?height=406&width=278&top_left_y=674&top_left_x=776)

Fig. 24

b) Answer: 5. Fix point $B$ at a distance of 3 from $P$ (Fig. 24). As point $A$ moves along a circle of radius 2 centered at $P$, vertex $C$ will move along a circle of radius 2, with its center $O$ located at a distance $O P=3$ from $P$ ( $\triangle O P B$ is equilateral). The farthest point on this circle from $P$ is at a distance $C O + O P = 5$.

$\nabla$ In the inequality to be proven, $P C \leqslant A P + B P$ (for any equilateral triangle $A B C$ and any point $P$), equality is achieved for all points $P$ on the arc $A B$ of the circumscribed circle of $\triangle A B C$, not containing point $C$.