$12 \cdot 27$ On the complex plane, four complex numbers are located at the four vertices of a square, three of which are $1+2 i, -2+i$, and $-1-2 i$. Then the fourth complex number is
(A) $2+i$.
(B) $2-i$.
(C) $1-2 i$.
(D) $-1+2 i$.
(E) $-2-i$.
(35th American High School Mathematics Examination, 1984)
[Solution] From the given, we know that the points $1+2i$ and $-1-2i$ are symmetric with respect to the origin, and their magnitudes (i.e., their distances to the origin) are both $\sqrt{5}$.

Therefore, the coordinate origin is the center of this square, from which we can determine that the fourth vertex is the point symmetric to $-2+i$ with respect to the origin, which is $2-i$.
Thus, the answer is $(B)$.