Parabola 10Given a line tangent to a parabola, along with the corresponding point of tangency, and given two other points on the curve, construct the parabola.  The solution to Parabola 10 is among the busiest constructions in this whole project. The proof of the construction is left to a separate page, Parabola 10 Support. Below, the reader will find the construction alone, bereft of explanation. The ConstructionBeginning with the objects as given in the Sketchpad document, below is the construction of the two solutions. The given objects are the points on curve P1, P2, and T, and the tangent line through T.  Construct A, the intersection of P1P2 and the tangent line; C, the midpoint of P1P2; G, the point on P1P2 such that AG is the geometric mean of AP1 and AP2; and H, such that AH divides P1P2 harmonically.  Construct point B, such that AGBT is a parallelogram. Draw the diagonals, which are highlighted here because they establish diameter directions for the two solutions. Let TH intersect AB at D. Dilate D with respect to point T, using scale factor 2, and let the image be E. Lines AT and AE are tangents to one of the solutions at points T and E.  Construct line CLK parallel to GT, meeting TB at L, and AT at K. Produce line TL to M, such that L is the midpoint of TM. Lines KT and KM are tangents to the second solution at points T and M.  For the parabola constructions, refer again to Parabola 5. In the interactive sketch below, the given objects can be dragged to confirm that the construction satisfies all conditions, and that it does not exist at all in cases that have no solution. |