Here is the problem: Given a point on the curve and the slope of the tangent at that point, and a second point on the curve construct (in the classical sense) additional points on the parabola...

I have created a Geogebra aplet to illustrate the constuction, enjoy...

(I got a note that some folks had trouble with the aplet opening correctly.. If you have Geogebra you can download my actual file here..)

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 paraboloid of revolution.. In the process he uses this theorem and quotes it as if it is well known... Yet it seems not to appear in texts then (1920) or now.

The basic principal is this: If from any point A on a parabola, Chords are drawn through any other two points B, and C, If vertical (in general lines parallel to the axis of symmetry) are drawn from B and C to intersect the opposite line AB or AC, the the line drawn to the two points of intersection will be parallel to the tangent line at A...

and if you want to read the construction... scroll down a little

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Given a point A, and a line tangent to it, t.. and a second point P, construct the parabola through A and P with the given tangent...

Through P draw a vertical line and select a point B on the vertical line

Draw lines AP and AB

Through AB draw a line,r, parallel to the tangent line, t.

Where r crosses AP, mark point D

Construct a vertical line, f, through point D

Where D crosses the line AB will be a point on the parabola....