SIMON-MARAIS 2023 A1

For $0 < q < 1$, the $q$-Fibonacci spiral is constructed as described below. An arc of radius 1 is first drawn inside a $1 \times 1$ square. A second arc is then drawn in a $q \times q$ square, then a third in a $q^2 \times q^2$ square, and so on ad infinitum, to create a continuous curve. The example on the left shows the case $q = \frac{1}{2}$, while the example on the right shows $q = \frac{\sqrt{5}-1}{2}$.

Prove that there exists a circle centred at the centre of the initial $1 \times 1$ square such that for each $0 < q < 1$, the limiting endpoint of the $q$-Fibonacci spiral lies on this circle.