Proof 1: Modulus 10
Proof 2: Modulus 5
I have a third proof idea... I'll see if I can flush it out later.
Are you a math professor/researcher? If not, where do you find / get inspiration for these?

(Hint: don't look for the curve, just examine the distance between them and how it changes with time)

I'll try (again) to find the equation for the path in a bit.
Edit: I thought I'd include the last equation I got, but kept failing to verify. So I don't know it if is correct.