(5) Let the function $f(x)$ be defined on $\mathbf{R}$, and consider the following three statements:
(1) The graph of the function $f(x)$ that satisfies the condition $f(x+2)+f(2-x)=4$ is symmetric about the point $(2, 2)$;
(2) The graph of the function $f(x)$ that satisfies the condition $f(x+2)=f(2-x)$ is symmetric about the line $x=2$;
(3) The graphs of the functions $f(x-2)$ and $f(-x+2)$ in the same coordinate system are symmetric about the line $x=2$.
The number of true statements is ( ).
(A) 0
(B) 1
(C) 2
(D) 3
5 Substitute $x-2$ for $x$ in
$$
f(x+2)+f(2-x)=4
$$

to get
$$
f(x)+f(4-x)=4 .
$$

If the point $(x, y)$ is on the graph of $y=f(x)$, then $4-y=f(4-x)$, meaning the point $(x, y)$'s symmetric point about $(2,2)$, which is $(4-x, 4-y)$, is also on the graph of $y=f(x)$. Conversely, this is also true, so (1) is a true statement. Substitute $x-2$ for $x$ in
$$
f(x+2)=f(2-x)
$$

to get $f(x)=f(4-x)$. If the point $(x, y)$ is on the graph of $y=f(x)$, then $y=f(4-x)$, meaning the point $(x, y)$'s symmetric point about the line $x=2$, which is $(4-x, y)$, is also on the graph of $y=f(x)$, so (2) is a true statement. Given that (2) is a true statement, it is not difficult to deduce that (3) is also a true statement. Therefore, the answer is D.