Geometric progression of length four on the circle

A sequence of rational points (P1, P2, P3,…) on a circle is said to constitute an r-geometric progression if either the sequence of their x-coordinates {a1, a2, a3,…} or their y-coordinates {b1, b2, b3,…} forms a geometric progression with common ratio r. This article examines a specific construction of such a progression, focusing on a sequence of four points located on a circle whose center lies along the x-axis. As a consequence, it is deduced that a certain family of unit circles whose centers lie on the x-axis cannot contain a rational sequence of length four in geometric progression of the form {1, r, r2, r3}.