I was recently re- reading through some old (1920) notes from the Philosophical Magazine that Dave Renfro sent me (THANKS, Dave) and came across a nice problem based on an old principal of parabolas known to Archimedes. The method, I learned in the article, was used by Archimedes in his On Floating Bodies, book two.. in the course of investigating the equilibrium of a floating parabaloid of revolution.. In the article the author gives this theorem and quotes it as if it is well known... Yet it seems not to appear in texts much then (1920) or now. I did find the question in An elementary treatise on pure geometry: with numerous example by John Wellesley Russell.
Here is the problem: Given a point on the curve A, and the slope of the tangent at that point, (AC) and a second point on the curve B, construct (in the classical sense) additional points on the parabola... C was place above B by chance, and can be anywhere along the tangent. I have placed the problem on a coordinate grid to present it as a function of x, but the actual coordinates of the points have no influence on the construction, although it is assumed that you know the direction of the axis of symmetry (in this case, vertical).
We begin by constructing a vertical line through B, and selecting a point D, somewhere along this line . Through this point draw another line parallel to the tangent and a second through point A. Finally draw a secant AB of the parabola.