For $0 < c < b < a$, let $(a+b-2 c) x^2+(b+c-2 a) x+(c+a-2 b)=0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements (I) If $\alpha \in(-1,0)$, then $b$ cannot be the geometric mean of $a$ and $c$ (II) If $\alpha \in(0,1)$, then $b$ may be the geometric mean of $a$ and $c$

Question

For $0 < c < b < a$, let $(a+b-2 c) x^2+(b+c-2 a) x+(c+a-2 b)=0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements

(I) If $\alpha \in(-1,0)$, then $b$ cannot be the geometric mean of $a$ and $c$

(II) If $\alpha \in(0,1)$, then $b$ may be the geometric mean of $a$ and $c$

only (II) is true

Both (I) and (II) are true

only (I) is true

Neither (I) nor (II) is true

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