11. Given an equilateral $\triangle A B C$, with its vertices arranged counterclockwise. Find the locus of the centroid of the equilateral $\triangle A^{\prime} B^{\prime} C^{\prime}$, where the vertices are also arranged counterclockwise, and points $A, B^{\prime}$, $C^{\prime}, A^{\prime}, B, C^{\prime}$, and $A^{\prime}, B^{\prime}, C$ are collinear respectively.
11. Angles are measured in the clockwise direction. Let $a, b, c$ be variable lines determined by the following conditions:
1. $A \in a, B \in b, C \in c$;
2. $\angle(a, b)=\angle(b, c)=\angle(c, a)$
$$
=\frac{\pi}{3} \text {; }
$$
3. $a, b$ and $c$ do not intersect at a single point.

The relations $A^{\prime} \in b \cap c, B^{\prime} \in a \cap c, C^{\prime} \in a \cap b$ determine a correspondence between all $\triangle A^{\prime} B^{\prime} C^{\prime}$ and all triples $(a, b, c)$. Let $a^{\prime}, b^{\prime}, c^{\prime}$ be the angle bisectors of $\angle(b, c), \angle(c, a), \angle(a, b)$, respectively. Then $a^{\prime}, b^{\prime}, c^{\prime}$ intersect at a point $X$, which is the incenter of $\triangle A^{\prime} B^{\prime} C^{\prime}$. Moreover,
$$
\begin{array}{l}
\text { (1) } \angle\left(a^{\prime}, b^{\prime}\right)=\angle\left(b^{\prime}, c^{\prime}\right)= \\
\angle\left(c^{\prime}, a^{\prime}\right)=\frac{2 \pi}{3} \text {. }
\end{array}
$$

Let points $K, L, M$ be such that $K B=K C, L A=L C, M A=M B, C_{1}, C_{2}, C_{3}$ are the circumcircles of triangles $C K B, A L C, B M A$ (see figure). Using the known fact
(・) If $P, Q$ are distinct points and $0<\alpha<\pi$, then the set $\{p \cap q$ : lines $p, q$ satisfy $P \in p, Q \in q$, $\angle(p, q)=\alpha\}$ is a circle passing through points $P, Q$; it follows that $C_{1}, C_{2}, C_{3}$ intersect at a point $P$ (the Torricelli point), and $A^{\prime}, B^{\prime}, C^{\prime}$ traverse $C_{1} \setminus \{P\}, C_{2} \setminus \{P\}, C_{3} \setminus \{P\}$, respectively, hence
(2) $K \in a^{\prime}, L \in b^{\prime}, M \in c^{\prime}$.
(3) $a^{\prime}, b^{\prime}, c^{\prime}$ take all directions except $K P, L P, M P$ (for example: if $K=P$, then $K P$ is the tangent direction of $C_{1}$ at $P$).

Let $S$ be the locus of point $X$, and let points $K, L, M$ be distinct. From (2), (1), (3), and (*), it follows that (4) $S=C \setminus \{P\}$.

Where $C$ is the circle passing through points $K, L$, and $M$, and $P \in C$.
Note: From (2), (1), (4), and (*), it follows that $\triangle K L M$ is equilateral (a famous theorem).
(Li Xin Nuan, translated)