1. $a$, $b$, $c$ are distinct positive numbers, $a^{2}+c^{2}=$ $2 b c$. Then which of the following relationships may hold? ( ).
(A) $a>b>c$
(B) $b>c>a$
(C) $b>a>c$
(D) $a>c>b$
-1.C.
If $a>b$, then $a^{2}+c^{2}>b^{2}+c^{2}>2bc$, which does not meet the condition, so we exclude options $(\mathrm{A})$ and $(\mathrm{D})$.

From $a^{2}-c^{2}=2c(b-c)$, we know that $a-c$ and $b-c$ have the same sign, thus we exclude option (B). Moreover, when $b>a>c$, $a^{2}+c^{2}=2bc$ might hold, for example, take $(a, b, c)=(3,5,1)$.