Tracing the path of the point A wheel of radius $R$ is rolling inside a fixed
circular cylinder of radius $2R$ as shown.
What is the trajectory followed by a point
on the rim of the wheel?

By observation, the only two points that seem to move in a straight line, are one at the centre of the cylinder and one at the common point of wheel and cylinder. I proved this too, using vector calculations. But I am not able to prove it for a general point. The answer given states that any general point on the rim of the wheel moves in a straight line. How to prove this?

This orbit is called hypotrochoid and has the parametric equations
$$
x(\theta) = (R-r)\cos\theta+r\cos\left(\frac{(R-r)\theta}{r}\right)\\
y(\theta) = (R-r)\sin\theta-r\sin\left(\frac{(R-r)\theta}{r}\right)\\
$$
in the present case we have $R=2R_0$ and $r = R_0$ so we have
$$
x(\theta) = R_0(\cos\theta+\cos\theta)\\
y(\theta) = R_0(\sin\theta-\sin\theta)\\
$$
so the parametric equations are
$$
x(\theta) = 2R_0\cos\theta\\
y(\theta) = 0
$$
In red the sought path inside the external circle (blue)

