Construction of Geometric Sequence in Lowest Terms/Porism

Proof
Apparent from the construction.

From Construction of Geometric Sequence in Lowest Terms, such a geometric sequence is of the form:


 * $P = \tuple {q^n, p q^{n - 1}, p^2 q^{n - 2}, \ldots, p^{n - 1} q, p^n}$

Thus when $n = 2$:
 * $P = \tuple {q^2, p q, p^2}$

and when $n = 3$:
 * $P = \tuple {q^3, p q^2, p^2 q, p^3}$