The problem is in section Definition in How to Solve It:
Construct the point of intersection of a given straight line and a parabola which the focus and the directrix are given.
This is a Straightedge and Compass Construction problem. Although it is impossible to construct a parabola by straightedge and compass, the points of intersection can be found:
Construct a point P on the given straight line c at equal distances from the given point F and the given straight line d.
However, the solution is not given in this book. I'd like to give a coordinate solution, which can be converted to straightedge and compass construction.
We put F onto the origin of Cartesian coordinates, then get the equations for P(x,y):
The solutions are:
Here let , then
can be constructed by Geometric Mean Theorem.
There is only a tangent point if q = 0, and no intersection or tangent points if q < 0.
If c is a vertical line, then the equation for P(x,y) (x is the distance from F to c) is . The solution is
.
Synthetic solutions can be found here.