One of the steps of deriving the equations for the parametric curve of a cycloid is the following:
Here we establish that the distance PT is equal to the distance OT, which then (alongside other steps) allows us to derive the parametric equation of the cycloid.
Every video or written proof I have seen on the proof for deriving the parametric curve consider this step to be "intuitive", but this does not strike to me as intuitive at all. Am I missing something obvious? Is there someway we could prove this fact? Is my intuition "broken"? (I hope not).
The image in the post was taken form this video: https://www.youtube.com/watch?v=wUDQFRZyE9Y&ab_channel=TimHodges (1:04)
As the circle rolls without slipping on the line the arc length from $P$ to $T$ matches the length along the $x$ axis. Imagine that the circle has a tape measure wound around it anchored at the origin that unrolls as the circle rolls.