See the Wikipedia page for satisfying proofs of the Cycloid's Involute, Arc Length, Area, and Pendulum properties. The involute property has a particularly neat proof, and it gives us a neat proof of the arc length property, and a shockingly elegant construction for the pendulum motion:
https://en.m.wikipedia.org/wiki/Cycloid
The article about cardioids has a very elegant proof of coffee cup caustic reflection property:
https://en.m.wikipedia.org/wiki/Cardioid