Can the lengths of the sides of a right-angled triangle be geometric?

Let a right-angled triangle have sides b, bq and bq², which form a geometric progression with the first term b and the denominator q. We will assume that b> 0, q> 1, and bq² is the hypotenuse. Let’s write down the Pythagorean theorem for a given triangle:

b² + (bq) ² = (bq²) ²;

b²q⁴ – b²q² – b² = 0;

q⁴ – q² – 1 = 0.

Replace q² = r> 0:

r² – r – 1 = 0.

r = (√5 + 1) / 2,

where

q = √ (√5 + 1) / √2.

Thus, the lengths of a right-angled triangle can be a geometric progression (with found q and any b).