140 THE MATHEMATICAL PRINCIPLES [BOOK I X IT that HA may be to IA as the rectangle un der a mean proportional between CG and GP, and a mean proportional between BH and HD is to a rectangle under a mean pro portional between GD and GB, and a mean proportional betweeen PI and 1C, and A will be the point of contact. For if HX, a par allel to the right line PI, cuts the trajectory in any points X and Y, the point A (by the properties of the conic sections) will come to be so placed, that HA2 will become to AP in a ratio that is compounded out of the ratio of the rec tangle XHY to the rectangle BHD, or of the rectangle CGP to the rec tangle DGB; and the ratio of the rectangle BHD to the rectangle PIC. But after the point of contac.t A is found, the trajectory will be described as in the first Case. Q.E.F. But the point A may be taken either between or without the points H and I, upon which account a twofold trajectory may be described. PROPOSITION XXIV. PROBLEM XVI. To describe a trajectory that shall pass through three given points, and touch two right lines given by position. Suppose HI, KL to be the given tangents and B, C, D, the given points. Through any two of those points, as B, D, draw the indefi nite right line BD meeting the tangents in the points H, K. Then likewise through any other two of these points, as C, D, draw the indefinite right line CD meeting the tan gents in the points I, L. Cut the lines drawn in R and S, so that HR may be to KR as the mean proportional between BH and HD is to the mean proportional between BK and KD ; and IS to LS as the mean pioportional between CI and ID is to the mean proportional between CL and LD. But you may cut, at pleasure, either within or between the points K and H, I and L, or without them ; then draw RS cutting the tangents in A and P, and A and P will be the points of contact. For if A and P are supposed to be the points of contact, situated anywhere else in the tangents, and through any of the points H, I, K, L, as I, situated in either tangent HI, a right line IY is drawn parallel to the other tangent KL, and meeting the curve in X and Y, and in that right line there be taken IZ equal to a mean pro portional between IX and IY, the rectangle XIY or IZ2, will (by the pro perties of the conic sections) be to LP2 as the rectangle CID is to the rect angle CLD, that is (by the construction), as SI is to SL2; and therefore