Let L and N be adjacent cusps of a cycloid.
Let P be a point on the cycloid between L and N.

Construct with straightedge and compass the tangent line to the cycloid at point P.

(In reply to Improved Solution by Harry)

There are several sites online giving instructions using only a straightedge and compass on how to construct a perpendicular from a point to a line and how to construct a perpendicular bisector using two points.
I applaud your approach, especially with that meeting the more stringent requirements. It is indeed an improved solution.

(Btw, I did not object to your offering involving constructing perpendicular lines. There are many sites that describe the process on how to draw them with straightedge and compass. I only objected to the omission of the steps you included in your improved solution):

In answer to how a perpendicular line may be drawn to a line from a point not on that line with a collasable compass:
Place one end of the compass on the point and draw an arc crossing the line at two points. Lift the compass and place one end on one of these crossed points and the other more than half the distance between it and the other crossed point, then draw an arc, repeating the process with the other crossed point such that the two new arcs cross forming a 'new' point. With the  straightedge, draw a line between the original point and the 'new' point, crossing the line. A perpendicular to the line is now drawn. 

Posted by Dej Mar on 2010-01-13 19:46:42