Length of Tangent to Astroid between Axes equals Radius of Stator

Theorem
Consider a circle $C_1$ of radius $b$ rolling without slipping around the inside of a circle $C_2$ of (larger) radius $a$ in a cartesian coordinate plane.

Consider a point $P$ on the circumference of $C_1$ where it is tangent to $C_2$ at point $A$ on the $x$-axis.

Consider the astroid $H$ traced out by the point $P$.

The segment of the tangent to $H$ between the $x$-axis and the $y$-axis is constant, immaterial of the point of tangency.

Proof

 * AstroidTangent.png

From the derivation of Equation of Astroid, $H$ can be expressed as:


 * $\begin{cases}

x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end{cases}$

Thus the slope of the tangent to $H$ at $\left({x, y}\right)$ is:

Thus the equation of the tangent to $H$ is given by:


 * $y - a \sin^3 \theta = -\tan \theta \left({x - a \cos^3 \theta}\right)$

The $x$-intercept is found by setting $y = 0$ and solving for $x$:

Similarly, the $y$-intercept is found by setting $x = 0$ and solving for $y$, which gives:
 * $y = a \sin \theta$.

The length of the part of the tangent to $H$ between the $x$-axis and the $y$-axis is given by:

which is constant.