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

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

Then $\left({a + b}\right)$, $\left({b + c}\right)$ and $\left({a + b}\right)$ 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:
 * $P = \left({p^2, p q, q^2}\right)$

Then:

Similarly:

Then: