sequences and series - Arithmetic progression & Geometric progression

If the $m$-th, $n$-th, and $p$-th terms of an A.p. and G.p. are equal and are $x,y,z$ respectively, prove that $x^{y-z}$. $y^{z-x}$. $z^{x-y}= 1$. To solve this question what I did is simply kept values of $x,y,z$ from G.P. (i.e. for $x = ar^{m-1}$ so on).
Can you help me to solve this in an more interesting way.