# Quotients of 3 Unequal Numbers are Unequal

## Theorem

Let $x, y, z \in \R_{\ne 0}$ be non-zero real numbers which are not all equal.

Then $\dfrac x y, \dfrac y z, \dfrac z x$ are also not all equal.

## Proof

Aiming for a contradiction, suppose $\dfrac x y = \dfrac y z = \dfrac z x$.

 $\displaystyle$  $\displaystyle \dfrac x y = \dfrac y z = \dfrac z x$ $\displaystyle \leadsto \ \$ $\displaystyle$  $\displaystyle x^2 z = y^2 x = z^2 y$ multiplying top and bottom by $x y z$ $\displaystyle \leadsto \ \$ $\displaystyle$  $\displaystyle x z = y^2, x y = z^2, y z = x^2$ $\displaystyle \leadsto \ \$ $\displaystyle$  $\displaystyle x y z = y^3, x y z = z^3, z y z = x^3$ $\displaystyle \leadsto \ \$ $\displaystyle$  $\displaystyle x^3 = y^3 = z^3$ $\displaystyle \leadsto \ \$ $\displaystyle$  $\displaystyle x = y = z$

This contradicts the assertion that $x, y, z$ are all unequal.

Hence the result by Proof by Contradiction.

$\blacksquare$