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Intuitively, why can $a, b$ cycle in ${\color{red}{b}} = \frac c{\color{red}{a}} \iff {\color{red}{a}} = \frac c{\color{red}{b}}$?

I'm NOT asking about algebra behind $ab = c \iff {\color{red}{b}} = \frac c{\color{red}{a}} \iff {\color{red}{a}} = \frac c{\color{red}{b}}.$ 1. Rather, **what's the intuition** why $\color{red}{a, b}$ can swap places, whilst c remains in the numerator?
2. **What's this phenomenon or behavior termed**? My son's teacher thinks this is called a _cyclic permutation_, but I want to double check because his teacher admitted he almost failed Abstract Algebra in his [BSc Mathematics Education](https://www.bu.edu/academics/wheelock/programs/mathematics-education/bs/).

I'm NOT asking about algebra behind $ab = c \iff {\color{red}{b}} = \frac c{\color{red}{a}} \iff {\color{red}{a}} = \frac c{\color{red}{b}}.$
1. Rather, what's the intuition why $\color{red}{a, b}$ can swap places, whilst c remains in the numerator?
2. What's this phenomenon or behavior called? A cyclic permutation?

arithmetic