Let $a,$ $b,$ $c,$ $x,$ $y,$ $z$ be nonzero complex numbers such that

Let $a,$ $b,$ $c,$ $x,$ $y,$ $z$ be nonzero complex numbers such that

\[a = \frac{b + c}{x - 2}, \quad b = \frac{a + c}{y - 2}, \quad c = \frac{a + b}{z - 2},\]and $xy + xz + yz = 5$ and $x + y + z = 3,$ find $xyz.$