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

Proof
From Tangent to Cycloid passes through Top of Generating Circle, the tangent to $C$ at a point $P = \left({x, y}\right)$ 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.