1. ## Astroid

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 $\displaystyle x=3t$.

2. ## Re: Astroid

Could you clarify how angle $\displaystyle x$ is defined? What determines the orientation of the lines that form this angle?

3. ## Re: Astroid

Originally Posted by corsica
Could you clarify how angle $\displaystyle x$ is defined? What determines the orientation of the lines that form this angle?
$\displaystyle x$ is the angle between the "horizontal through the centre of the small circle" and the "line joining the centre of the small circle and the red dot". See Astroid.

4. ## Re: Astroid

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