Given a circle with diameter AC and points A, B, C on the circumference such that AC = BC = 2q and AB = q, find the value of q.

This problem contains contradictory information, which makes it impossible to solve as stated.

Here's why:

1. The problem states "AC is the diameter of the circle with centre O". A fundamental theorem in geometry states that the angle subtended by a diameter at any point on the circumference is a right angle. Therefore, $\angle ABC$ must be $90^\circ$.
2. The problem also states "$AC = BC = 2q$ and $AB = q$". The handwritten notes on the diagram interpret $2q=8$ and $q=4$, leading to $AC=8$, $BC=8$, and $AB=4$.
3. If $\triangle ABC$ is a right-angled triangle at B (due to AC