What is the proof that the motion of the moon around the Sun is cycloidal orbital path?

Answer 1

If you construct a scale drawing of the path of the earth (a circle is close enough given the low eccentricity of it's path around the sun) and then plot the path of the moon properly you will see that the orbital path of the earth forces the moons orbit to be sort of a wavy sine curve that is concave toward the sun. So, the proof is mathematical, specifically graphic.

In effect the moon orbits the sun in an orbit highly perturbed by the presence of the Earth.

Answer 2

I think you'd have to do the maths. I've seen lots of statements that say the moon's orbit is always concave with respect to the sun, but I've never demanded proof.