### Suppose $a \ne 0, b \ne 0, c \ne 0$ and $\dfrac{ a } { b } = \dfrac{ b } { c } = \dfrac{ c } { a } .$ Find the value of $\dfrac{ a - b + c } {a + b - c }.$

$1$
1. We need to find the value of $\dfrac{ a - b + c }{ a + b - c } .$
Let $\dfrac{ a } { b } = \dfrac{ b } { c } = \dfrac{ c } { a } = k$
2. From $\dfrac{ a } { b } = \dfrac{ b } { c } = \dfrac{ c } { a } = k,$ we get
\begin{align} & a = bk, b = ck, \text{ and } c = ak \\ \implies & a = ak^3 \\ \implies & k^3 = 1 \\ \implies & k = 1 \\ \implies & a = b = c \end{align}
$$\dfrac{ a - b + c } {a + b - c } = \dfrac{ a - a + a } { a + a - a } = 1$$