Let fall the perpendicular PH on DX, and from K draw KN parallel to PF, meeting DF, produced if necessary, in N. Because of the equiangular triangles DKN and DPF, KN is to KD, as PF is to PD (4. 6 Ev.) ; whence, PF being equal to PD (15. 1 Sup.), KN is equal to KD. And since KN is to KD, as PF is to PD, and, because of the equiangular triangles DKX and DPH, DK is to KX, as DP to PH (4. 6 Eu.), by equality, KN is to KX, as PF to PH (22. 5 Eu.), or in the determining ratio ; therefore KN or KD is equal to KM (Hyp. and 9. 5 Eu). From the centre K, at the distance KD, KN or KM, let the circle DNM be described ; and, because of the parallels PF and KN, the rectangle DPK is to the rectangle DFN, or, which is equal (Cor. 3. 36. 3 Eu.), the difference of the squares of KF and KM, as the square of DP is to the square of DF (20. 6 and Cor. 3. 22. 5 Eu). Cor. The segment (KD), of a right line (DM) parallel to an asymptote (CZ) of a hyperbola, between any point (K) in the parallel, and the directrix, is to a perpendicular (KX) let fall from the same point on the directrix, in the determining ratio. For, because of the equiangular triangles DKX and DPH, KD is to KX, as PD, or its equal (15. 1 Sup.) PF, is to PH (4. 6 Eu.),or in the determining ratio. Scholium. The reasoning in this proposition and corollary applies, whether the point taken in the parallel, be within or without the hyperbola, as the point k, a right line being supposed to be drawn from k to F, by substituting the small letters k, x, m and n for their respective capitals. PROP. XVII. THEOR. If two right lines, parallel to each other, both touch or both cut in two points, or one of them touch, and the other so cut, a hyperbola or opposite hyperbolas, and meet a right line parallel to an asymptote ; the squares of the segments of the tangents, or rectangles under the segments of the secants, between the right line parallel to the asymptote, and the point or points wherein they meet the hyperbola or hyperbolas, are to each other, as the segments of the right line parallel to he asymptote, between the parallels, and the concourse of that right line, with the hyperbola, which it meets. Let, in fig. 1, RL and rl be two tangents, parallel to each other, touching opposite hyperbolas in R and r, meeting a directrix DG, in H and h, and DK parallel to the asymptote CZ in L and l; and let secants ST and st meet a hyperbola, as in fig. 1, or opposite hyperbolas, as in fig. 2, in S and T, s and t, the directrix DG in G and y, and the parallel to the asymptote in K and k. The squares of LR and Ir in fig. 1, and the rectangle SKT and skt in fig. 1 and 2, are to each other, as the right lines PL and Pl, PK and Pk. Join KF and DF; and KD and kD in fig. 1 and 2, and LD and ID in fig. 1, have to perpendiculars let fall from the points K and k, L and l, on the directrix DG, the determining ratio (Cor. 16. 1 Sup.), and let, first, the parallel right lines which meet DK, be, in fig. 1, the tangent LR and the secant ST, and since the rectangle SKT is to the difference of the squares of KF and KD, as the square of KG is to the difference of the squares of KG and KD (13. 1 and Cor. 16. 1 Sup.), and the difference of the squares of KF and KD is to the rectangle DPK, as the square of DF is to the square of DP (16. 1 Sup. and Theor. 3. 15. 5. Eu.), by compounding these ratios, the rectangle SKT is to the rectangle DPK, in a ratio compounded of the ratios of the rectangle SKT to the difference of the squares of KF and KD, and of the same difference to the rectangle DPK, (Def. 13. 5 Eu.), or, which has been just shewn to be equal, of the ratios of the square of KG to the difference of the squares of KG and KD, and of the square of DF to the square of DP, In like manner it may be proved, that the square of RL is to the rectangle DPL, in a ratio, compounded of the ratios of the square of LH to the difference of the squares of LH and LD, and of the square of DF to the square of DP. But in these two compounded ratios, that of the square of DF to the square of DP is common to both, and, because of the equiangular triangles KGD and LHD, the ratios of KG to KD and of LH to LD are cgual (4. 6. Eu.), and therefore those, of the square of KG to the square of KD, and of the square of LH to to the square of LD (20. 6. and cor. 3. 22. 5. Eu.), and therefore those, of the square of KG to the difference of the squares of KG and KD, and of the square LH to the difference of the squares of LH and LD (Schol. 18. 5 Eu.) therefore these compound ratios being compounded of equal ratios, are equal (22. 5. Eu.), therefore the ratios equal to them of the rectangle SKT to the rectangle DPK, and of the square of RL to the rectangle DPL are equal, and, by alternating, the rectangle SKT is to the square of RL, as the rectangle DPK, is to the rectangle DPL (16. 5 Eu.), or, the side DP being common to both rectangles, as PK is to PL (1. 6. Eu). In like manner, if, instead of the tangent RL and secant SKT, the tangent rl and secant skt be used, the truth of the proposition may be shewn, a right line being supposed to be drawn from k to F, by substituting the small letters s, t, k, g, r, 1 and h for the corresponding capitals, as is manifest. And, by a similar reasoning, it may be proved, in both figures, that the rectangles EKT and skt, are to cach other, as the seg. ments KP and kP, of the right line parallel to the asymptote, between the parallels ST and st, and the point P. PROP. XVIII. THEOR. If a right line, touching a hyperbola, or cutting a hyperbola or op posite hyperbolas in two points, meet two right lines parallel to an asymptote; the squares of the segments of the tangent, or rectangles under the segments of the secant, between the parallels, and the point or points, wherein the tangent or secant meets the section or sections, are to each other, in a ratio, compounded of the ratios, of the segments of the parallels, between the tangent or secant, and the points wherein the same parallels meet the hy. perbola, and of the segments of the same parallels, between a right line passing through the focus adjacent to them, and perpendicularly cutling them, and the adjacent directrix. Let a right line XZ, see fig. 1, touching a hyperbola in R, or ST, see fig. 1 and 2, cutting a hyperbola or opposite hyperbolas in S and T, meet two right lines DK and GL, parallel to an asymptote CH, and meeting the adjacent directrix DG in D and G; from the focus F adjacent to the parallels, let the right line FBA be drawn, perpendicularly meeting these parallels in A and B, the squares of XR and ZR, or rectangles SKT and SLT, as the case may be, are to each other in a ratio compounded of the ratios of XP to ZQ or PK to QL, and of DA to GB. First, the rectangles SKT and SLT, see fig. 1 and 2, are to each other, in a ratio compounded of the ratios of PK to QL and of DA to GB. Join FD, FK, FL and FP ; let U and u be the points in which ST and XZ meet the directrix DG ; take AV equal to AD), and through K draw KM parallel to PF, meeting DÈ produced in M. The right line FV is cqual to FD (4. 1 Eu.), and PF to PD (15. 1 Sup.) ; therefore the triangles DFV and DPF are isosceles, and,'having the angle PDF common, are cquiangular, therefore DV is to DF, as DF is to DP (4. 6 Eu.), or, which is equal, because of the parallels PF and KM, as FM is to PK ; (2. 6 and 16. 5 Eu.), therefore the rectangle under DV or twice DA and PK, is equal to the rectangle DFM (16. 6 Eu.): but, because of PF equal to PD, and parallel to KM, the right lines KD and KM are equal, and a circle described from the centre K, at the distance KD, would pass through M, and so the rectangle DFM is equal to the difference of the squares of KD and KÉ (Cor. 3. 36. 3 Eu.) ; therefore the rectangle under twice DA and PK is equal to the difference of the squares of KD and KF. In like manner, if BN be taken equal to BG, and FQ, FG and FN be joined, and GF produced meet a right line drawn through L parallel to QF, it may be proved, that the rectangle under twice GB and QL is equal to the difference of the squares of LG and LF. But the ratios of KD and LG, to perpendiculars let fall from K and L on DG, are equal to the determining ratio (Cor. 16. 1 Sup.), therefore the rectangle SKT is to the difference of the squares of KD and KF, as the square of KU is to the difference of the squares of KU and KD (13. 1 Sup.); for a like reason, the rectangle SLT is to the difference of the squares of LG and LF, as the square of LU is to the difference of the squares of LU and LG; but, because of the equiangular triangles KUD and LUG, the ratios of KU to KD and of LU to LG are equal (4. 6 Eu.), and therefore those of the square of KU to the square of KD, and of the square of LU to the square of LG (20. 6 and Cor. 3. 22. 5 Er.), and therefore those, of the square of KU to the difference of the squares of KU and KD, and of the square of LU to the difference of the squares of LU and LG (Schol. 18. 5 Eu.); therefore the ratios of the rectangle SKT to the difference of the squares of KD and KF, and of the rectangle SLT to the differ.ence of the squares of LG and LF, which are equal to these |