Normal to Cycloid passes through Bottom of Generating Circle/Proof 2

Theorem

Let $C$ be a cycloid generated by the equations:

$x = a \paren {\theta - \sin \theta}$

$y = a \paren {1 - \cos \theta}$

Then the normal to $C$ at a point $P$ on $C$ passes through the bottom of the generating circle of $C$.

Proof

From Tangent to Cycloid passes through Top of Generating Circle, the tangent to $C$ at a point $P = \tuple {x, y}$ passes through the top of the generating circle.

By definition, the normal to $C$ at $P$ is perpendicular to the tangent to $C$ at $P$.

From Thales' Theorem, the normal, the tangent and the diameter of the generating circle form a right triangle.

Thus the normal to $C$ at $P$ meets the generating circle at the opposite end of the diameter to the tangent.

Hence the result.

$\blacksquare$

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid: Problem $3$