The points A(0, 0), B(8, 0) and C(8, 6) lie on a circle…

Step 1. The diameter. $\angle ABC = 90^\circ$, since $AB$ is horizontal and $BC$ is vertical. An angle of $90^\circ$ at the circle stands on a diameter. So $AC$ is a diameter.

Step 2. The centre. It is the middle of $A(0, 0)$ and $C(8, 6)$: $(4, 3)$.

Step 3. The radius. The length of $AC$ is $\sqrt{64 + 36} = 10$. So the radius is $5$.

Step 4. The two points. For $D(0, 6)$: the distance from $(4, 3)$ is $\sqrt{16 + 9} = 5$. It is on the circle; $ABCD$ is a rectangle. For $E(2, 8)$: the distance is $\sqrt{4 + 25} = \sqrt{29}$, which is more than $5$. It is not on the circle.