Normal to Cycloid passes through Bottom of Generating Circle/Proof 1

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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 Normal to Cycloid, the equation for the normal to $C$ at a point $P = \tuple {x, y}$ is given by:




Sources