There isn't much difference between a catenary and its approximate parabola.
The catenary is the correct model of an ideal chain.
But the parabola fits fairly well too.
The polar point of a line in a parabola is a common point to the chords defined by the common tangents through the points on the line.
(Go to Full Screen if the green points won’t drag)
The locus of the intersection of tangents at the ends of chords of a parabola through a given fixed point is a straight line.
This is called the polar line of the given point
We use a trick to let the trace "open up" as you drag a point.
The trick is this: an initial point is given parametric location s*t, create a tangent at this point and its envelope as s varies.
Now hide the original point and create another point with parameter t, and make it draggable.
Dragging the new point changes the value of t and we see a trace from 0 to t.