12. Let $S$ be a set of points in the Cartesian coordinate system, satisfying that two of the quantities $3$, $x+2$, $y-4$ are equal and the third quantity is not greater than the other two. Which of the following statements about the point set $S$ is correct? ( ).
(A) a single point
(B) two intersecting lines
(C) three lines intersecting at three points
(D) a triangle
(E) three rays with a common endpoint
12. E.

Case analysis.
When $3=x+2 \geqslant y-4$,
$$
S=\{(x, y) \mid x=1, y \leqslant 7\}
$$

it represents a ray perpendicular to the $x$-axis with an endpoint at $(1,7)$;
When $3 \leqslant x+2=y-4$,
$$
S=\{(x, y) \mid x \geqslant 1, y=x+6\}
$$

it represents a ray with an endpoint at $(1,7)$ and a slope of 1;
When $3=y-4 \geqslant x+2$,
$$
S=\{(x, y) \mid x \leqslant 1, y=7\}
$$

it represents a ray perpendicular to the $y$-axis with an endpoint at $(1,7)$.