General 2Given two tangent lines (tangent1 and tangent2tangent2) with their corresponding points of tangency (T1 and T2) and one other point (P) on curve, construct the conic section.  If two more points on curve can be produced, this construction will be practically finished. Let the given tangent lines intersect at A. Construct line T1T2, and let B be the midpoint of line segment T1T2. This makes AB a diameter of the curve (Conica (II, 29)). It bisects chord T1T2, so it must also bisect any chord parallel to T1T2. Construct point C such that PC is parallel to T1T2 and is bisected by AB.  Construct lines CB and AP, letting them intersect at D. Let line AP intersect line T1T2 at E. It is left as an exercise to show that PD divides AE harmonically. As a result, point D also is on the curve (Conica (III, 37).  The figure now has points on curve T1, T2, P, C, and D. See Newton and Conic Parameters for completion of the construction of the conic section through five given points. There are shortcuts to be found if you watch for them. Drag the given objects in the websketch below, and confirm that the construction holds up for every positioning. |