12. Let the family of lines $M$ be:
$x \cos \theta + (y-2) \sin \theta = 1 (0 \leqslant \theta \leqslant 2 \pi)$.
For the following four statements:
(1) All lines in the family $M$ pass through a fixed point;
(2) There exists a fixed point $P$ that does not lie on any line in the family $M$;
(3) For any integer $n (n \geqslant 3)$, there exists a regular $n$-sided polygon, all of whose sides lie on lines in the family $M$;
(4) The triangles formed by the lines in the family $M$ all have equal area,
Among these, the true statements are $\qquad$ (write down the numbers of all true statements).
12. (2)(3).

From $x \cos \theta+(y-2) \sin \theta=1$, we know that the distance from point $P(0,2)$ to each line in the line system $M$ is
$$
d=\frac{1}{\sqrt{\cos ^{2} \theta+\sin ^{2} \theta}}=1,
$$

which means the line system $M$ consists of all the tangents to the circle $\odot C: x^{2}+(y-2)^{2}=1$. Therefore, there exist two parallel lines in the line system $M$. Hence, statement (1) is incorrect.

Since point $(0,2)$ does not lie on any line, statement (2) is correct.

For any $n \geqslant 3$, there exists a regular $n$-sided polygon whose incircle is $\odot C$, so statement (3) is correct.

In the line system $M$, there can be two triangles of different sizes, $\triangle A B C$ and $\triangle A E F$, so statement (4) is incorrect.
In summary, the correct statements are (2)(3).