Parabola 12Given four tangent lines, construct the parabola and the four points of tangency.  This follows from the challenge General 4, and may be considered a limiting case of that one. That is how this solution was first derived. However, the proof was simply inadequate, more like hand waving than anything else. Newton’s proof for the general case required certain conditions that cannot be filled by a parabola. It specifically required parallel tangent lines and a center. Proof of the construction below can be found in Parabola 12 Support. For the general conic, five tangent lines were required. In the parabola case, however, only four are needed. They are given as the produced sides of quadrilateral ABCD, which is actually an anti‑quadrilateral here. The general case began with the construction of two diameters, so that the center could be located at their intersection. A parabola has no center, but construction of a single diameter gives the direction for all others. It is constructed by the same procedure used in General 4.  Although ABCD is here warped into an anti‑quadrilateral, AC and BD may still be regarded as its diagonals. Construct the line joining the midpoints of the diagonals. It is shown here as the brown line. In Principia, Book I, Lemma 25, Newton concluded that this line would pass through the center of the section, as do all diameters. He was referring to ellipses, circles, and hyperbolas. A parabola has no center, but regardless, this is a diameter.  From here we move on to the point of tangency for line AD. In the quadrilateral, AD is adjacent to AB and CD, so disregard BC for now. Construct the line through A parallel to CD, and the line through D parallel to AB. These two lines meet at K, which lies on the diameter through the required point of tangency. Construct the line through K parallel to the previously constructed diameter. The new diameter intersects AD at L, the point of tangency.  That last part is repeated to construct the points of tangency on all four tangent lines. The sketch now has four tangent lines and the corresponding point of tangency for each. Only two of them are required for construction of the parabola. See Parabola 5 for construction of the parabola given only two tangencies. The parabola in the Web Sketch below is controlled by tangent intersection points A, B, C, and D, the purple points. Drag those to change to parabola. |