If $a^{x} = b$ , $b^{y} = c$ and $c^{x} = a$ , prove that: $x y z = 1$ .

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a^{x} = b, b^{y} = c, c^{z} = a

x y z = 1

\frac{(x - b) (x - c)}{(a - b) (a - c)} + \frac{(x - c) (x - a)}{(b - c) (b - a)} + \frac{(x - a) (x - b)}{(c - a) (c - b)} = 1

a x + b y + c = 0, b x + c y + a = 0

c x + a y + b = 0

a^{3} + b^{3} + c^{3} = 3 a b c

a + b + c = 0

If a, b, c are in A.P., prove that the straight lines $a x + 2 y + 1 = 0, b x + 3 y + 1 = 0$ and $c x + 4 y + 1 = 0$ are concurrent.

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If ax=b, by=c and cx=a, prove that: xyz=1.