Sum of Pair of Elements of Geometric Sequence with Three Elements in Lowest Terms is Coprime to other Element

Theorem
Let $P = \tuple {a, b, c}$ be a geometric progression of integers in its lowest terms.

Then $\paren {a + b}$, $\paren {b + c}$ and $\paren {a + c}$ are all coprime to each of $a$, $b$ and $c$.

Proof
Let the common ratio of $P$ in canonical form be $\dfrac q p$.

By Form of Geometric Progression of Integers in Lowest Terms:
 * $P = \tuple {p^2, p q, q^2}$

Then:

Similarly:

Then: