Intersection of rotating lines in TikZ

I'm doing a presentation on Beamer about Maclaurin's trissectriz and I want to show the intersection point between the two rotating lines. Here's my code so far.

The TikZ library intersections can be used to calculate intersection coordinates between paths:



Different values for the initial parameter i value as well as for the increment (-4 instead of 5) have been chosen. This is to avoid the construction lines to coincide (which would imply an infinite number of intersections).