Refer to the attached figure.
The ratio of the radii of the circles is 4:1. Angles $\displaystyle x$ and $\displaystyle t$ are shown. Prove that $\displaystyle x=3t$.
first, note the extension of the segment forming angle t in the diagram.
for the small circle, consider the arclength from the point of tangency of the two circles to the red dot ...
s = 1(x+t)
for the large circle, the arclength from the same point of tangency to (4,0) is ...
s = 4t
the two arclengths are the same since the small circle "rolls" inside the larger.
4t = x+t
x = 3t