but FD.DK =DA; hence DM2 =DA?, or DM=DA, and M must coincide with A. Cor. 1.-If a straight line be drawn through any point within the parabola, not parallel to a diameter, it will meet the curve in two points. For, if the line is not parallel to the directrix, it will meet it if produced, and must therefore cut the curve in one point, as H ; and hence may be found the points D, E, A, and therefore also the other point of intersection, as G. CoR. 2.-At the same point, in the curve, there cannot . be more than one tangent. Cor. 3.-A tangent bisects the angle formed by a straight : line drawn to the focus, and another perpendicular to the directrix from the point of contact ; it also bisects, and is perpendicular to the line that subtends that angle (Cor. 2, and I. 3). Cor. 4.- If a straight line from the focus be perpendi cular to a cord, it will bisect that part of the directrix which is intercepted by perpendiculars falling upon it from the extremities of the cord; and conversely. If FL be perpendicular to GH, it will bisect AE. For the circle EKF being described about the centre H, FL = LK, hence GK = GF, but GF=GA; therefore G is the centre of the circle AKF, and DA? = FD:DK=DEP. PROPOSITION VI. A straight line terminated by the parabola, and parallel to a tangent, is an ordinate to the diameter that passes through the point of contact. The cord GH, parallel to A D E c в the tangent MQ, will be bisected by the diameter MN. For, let GA, HE, be perpendicular to AB, and let FD be perpendicular to and meet MQ, GH, in Q, L. Then, because FD is perpendicular to MQ, it is also per- / pendicular to GH, and there G Cor. 1.-—The square of a perpendicular, upon any diameter, from a point in the curve, is equal to the rectangle under the parameter of the axis, and the absciss corresponding to the ordinate from the same point. For it was proved that GK2 = 4CV · MN. Cor. 2. If there be two diameters, and from the vertex of each a semi-ordinate be applied to the other, the ab scisses will be equal. For, let DK, and EH, produced (figure to proposition VI.) be the diameters; then ĜH is an ordinate to DK, and one from M applied to EH will be the other. Let the perpendicular GK upon DK be called P; and that from M upon EH, P'; the abscissa MN, A; and that of EH, A'. Then (Cor. 1) 4A · CV=P?, and 4A'. CV = P2 ; but since GN =NH, the perpendiculars P and P' are equal (Pl. Ge. I. 26); hence P2 = P2; and therefore 4A CV=4A'. CV, or A = A'. Cor. 3.-The square of that part of a tangent, between : the point of contact and any diameter, is equal to the rectangle under the external segment of that diameter, and the parameter of the diameter which passes through the point of contact. For RM=GN, and RG=MN (figure to proposition VI.) Cor. 4.-The squares of ordinates, or semi-ordinates, to any diameter, are to one another as their corresponding abscisses ; and the squares of perpendiculars, from the same points, are in the same ratio. Let 0, O', be two semi-ordinates to the same diameter; A, A', the corresponding abscisses, and P the parameter of the diameter. Then 02 =P: A, and 0'2=P: A'; therefore 02 : 02=P:A:P:A'=A:A' (Pl. Ge. VI. 1). Cor. 5.-If the squares of parallel lines, drawn from cer tain points, to meet a line given in position, be to one another as the parts they cut off towards one extremity, these points will be in the curve of a parabola, which has the given line for a diameter. For if the parallel lines be considered as ordinates, and the segments cut off from the given line as abscissæ, it follows (Cor. 4) that the given line is a diameter to a parabola passing through the extremities of the ordinates. PROPOSITION VIII. A subtangent, upon any diameter, is bisected in the vertex of that diameter. Let the tangent GM meet any diameter VN in M, and let GN be an ordinate to it, from the point of contact, the subtangent MN is bisected in V (figure to proposition VII.) For, let the diameter GL, and its semi-ordinate VL, be drawn, then is the absciss GL=VN (1. 7, Cor. 2); but, since LM is a parallelogram (I. 6, Cor. 1), GL = MV; therefore MV=VN. · Cor. 1.—That part of the axis, between the focus and any tangent, is of the parameter of the diameter pass ing through the point of contact. Cor. 2.- If MV = VN, and GN a semi-ordinate, then GM is a tangent, or, if GM be a tangent, GN is a semi-ordinate. PROPOSITION IX. That ordinate of a diameter which passes through the focus, is equal to its parameter. · Let GE be any diameter, and RE the semi-ordinate to it, which passes through the focus; then 2RE = 4GD (figure to proposition VII). For RE = 4GD · GE (I. 7); but GE=FM =GD (I. 8, Cor. 1). Therefore RE = 4GD, RE = 2GD, and ARE = 4GD. · CoR.—If an ordinate to any diameter pass through the focus, the absciss will be equal to the distance of the vertex from the focus.. . PROPOSITION X. If from any point in the parabola, a parallel to a diameter be drawn to meet an ordinate to the same, the rectangle under the parameter of the diameter and the parallel will be equal to the rectangle under the segments of the ordinate. From any point H, in the curve, let HE be drawn parallel to the diameter VN, to meet its ordinate GQ in E; then P: HE=GE: EQ. ** ind L 1110g ... E For, let the semi-ordinate HK be drawn; then GN = any diameter is to the to the difference of their · abscisses. For P. HE = EG · EQ = (GN + HK) (GN - HK). Therefore (Pl. Ge. VI. 16), P:GN + HK =GN - HK: HE, and HE=VN - VK. Cor. 2.-Straight lines, drawn parallel to a diameter, from points in the curve, to meet any cord, are to one another as the rectangles under the segments of the cord. For let P be the parameter of the diameter ; L, L', two of the parallel lines; and R, R', the rectangles under the corresponding segments of the cord, then PL=R, P.L' = R', therefore P·L:P.L'=R:R', or (Pl. Ge. VI. 1) L:L' =R:R'. Cor. 3.-If two parallel cords meet any diameter, the rectangles under their segments will be to each other directly as the parts of the diameter which they interFi' cept from the vertex. For Let S and S' be the segments cut off from the diameter; R and R' the rectangles under the segments of the cords; and P the parameter of the diameter to which the cords are ordinates. Then P.S= R, and P.S=R'; therefore R:R' =P:S:P.S=S:S. PROPOSITION XI. If a diameter be cut by any straight line passing through two points in the parabola, the part intercepted from the vertex will be a mean proportional between the abscisses corresponding to the two ordinates drawn from the same points in the curve. |