Quick Proofs of Properties of the Cycloid

I. Normal chords

II. Normal cycloids

III. Horizontal and normal velocities

IV. Brachistochrone

V. Self-evolute / self-involute

VI. Total and partial areas

VII. Total and partial arc lengths

VIII. Tautochrone (isochrone)

IX. Generating circle of brachistochrone

X. Parametric equations [at last]

XI. Electromagnetic interpretation.

P.S.: For a shorter version of this article, concentrating on the brachistochrone and adding an "optical interpretation", see "The cycloid as brachistochrone: A one-page proof, from first principles, without calculus", arxiv.org/html/2507.22548v1, 30 July 2025.