PROP. LXV. THEOR.

two right lines, touching a conick section or opposite sections, meet each other, and from any point in one of the tangents, two right lines be drawn, one parallel to the other tangent, to the right lines joining the contacts, and the other in any manner, cutting in two points the section or sections; the square of the right line drawn to that joining the contacts, is to the rectangle under the segments of the secant, between the tangent and the section or sections, in the given ratio of the squares of the segments of tangents, or rectangles under the segments of secants, parallel to these right lines, between their concourse, and the section or sections.

Let PK and QK, see all the fig. of the prec, prop, touching a conick section or opposite sections in P and Q, meet each other in K, and from any point R, in one of the tangents KPR, let two right lines RO and RST be drawn, one RO parallel to the other tangent QK, to the right QPO joining the contacts, and the other RST in any manner, cutting the section or sections in S and T; the square of RO and the rectangle SRT are to each other, in the given ratio of the squares of the segments of tangents, or rectangles under the segments of secants, parallel to RO and RT, between their concourse, and the section or sections.

From any point D in the tangent PK, draw right lines DH and DZ parallel to RO and RT, cutting the section or sections in G and H, X and Z, and let DH parallel to RO, meet the right line joining the contacts in L; the rectangle SRT is to the rectangle XDZ, as the square of PR is to the square of PD (Cor. 1. 14. 1 Sup.), or, because of the equiangular triangles PRO and PDL, as the square of RO is to the square of DL (4 and 22. 6 and 16. 5 Eu.), or its equal (64. 1 Sup.), the rectangle GDH; therefore, by alternating and inverting, the square of RO is to the rectangle SRT, as the rectangle GDH is to the rectangle XDZ (16. 5 and Theor. 3. 15. 5 Eu.), or in the given ratio of the squares of the segments of tangents, or rectangles under the segments of the secants, parallel to RO and RT, between their concourse and the section or sections (42. 1 Sup).

PROP. LXVI. THEOR.

If through the concourse (K), of two right lines (PK and QK), touching a conick section or opposite sections, there be drawn a right line, meeting the section or sections in two points, and the right line (PQ) joining the contacts; the right line so drawn is cut harmonically in the concourse of the tangents, the points in which it meets the section, and that, in which it meets the right line joining the contacts.

Case 1. Let the right line drawn through K, see both fig. as KTS, not be a diameter, and through T and S let BTE and NSO be drawn parallel to PQ, meeting the tangents in B and E, N and O, and the section or sections in T and I, S and U, through K let the diameter KH be drawn (Cor. 35. 1 Sup.), meeting the right lines BE, PQ and NO in X, H and Z, and, because it bisects the right line PQ in H (Cor. 1. 49 1. Sup.), it bisects BE and NO in X and Z (4. 6 and 22. 5 Eu.), and because TI and SU, terminated by the section or sections, are parallel to PQ, they are bisected in X and Z (Cor. 1. 32. 1 Sup.); therefore the segments EI and TB and the segments OU and SN are equal, and the rectangles BTE and NSO severally equal to the rectangles TBI and SNU.

And because of the parallels BE and NO, the right line NS is to BT, as SO to TE, therefore the rectangles NSO and BTE are similar; and therefore these rectangles, or the rectangles SNU and TBI are to each other, as the squares of NS and BT

(22. 6 Eu.), or, because of the equiangular triangles KNS and KBT, as the squares of KN and KB; but the same rectangles SNU and TBI are to each other, as the squares of PN and PB (Cor. 1. 14. 1 Sup.); therefore the squares of KN and KB are to each other, as the squares of PN and PB (11. 5 Eu.), and therefore KN is to KB, as PN is to PB (22. 6 Eu.), and of course, because of the parallels, KS is to KT, as YS is to 'TY; therefore the right line KS is cut harmonically in the points K, T, Y and S (Def. 24. 1 Sup).

Case 2. When the right lines PK and QK (see fig. 1) touch the same section, and the right line KGD drawn through K, is a diameter, let it meet PQ in H, and the section or opposite sections in G and D; the right lines GL and DR drawn through these points parallel to PQ are tangents (Cor. 1.49. 1, 32. 1 and Def. 12. 1 Sup.); let them meet the tangent KR in L and R; and because of the parallels, KR is to KL, as DR to GL, or which is equal (Cor. 2. 14. 1 Sup. and 22. 6 Eu.), as PR to PL; therefore, because of the parallels, KD is KG, as HD is to GH, and of course, the diameter drawn through K, is cut harmonically in the points K, G, H and D (Def. 24. 1 Sup).

Cor. From the demonstration of this proposition, it follows, that a tangent (KR), which meets two parallel tangents (GL and DR) and the right line (DG) joining their contacts, is cut harmonically in the contact (P) and the points (R, L and K), in which it meets the tangents, and the right line KGD joining their contacts.

Schol. The reasoning in case 2 and the cor. applies to fig. 2, 48, 1 Sup.

PROP. LXVII. THEOR.

Each of three tangents to a conick section or opposite sections, which meets the other two, and the right line joining their contacts, is cut harmonically, in the point of contact, and the points, in which it meets the other tangents, and the right line joining their con

tacts,

If two of the tangents be parallel, and the third meet the right line joining their contacts, the proposition is manifest from the preceding corollary.

But if the three right lines PK, QK and GH, touching the conick section or opposite sections in P, Q and T, meet each other in K, G and H, and PQ joining the contacts of two of them PK and QK, meet the third GH, produced if necessary, as in D; DH is to DG, as TH to TG.

Through H, the intersection of the tangents TH and QH, let a right line HRS be drawn, parallel

to the other tangent KP, meeting the section or each section in R and S, and PQ in L; the square of HL is equal to the rectangle RHS (64. 1 Sup.); but, because of the parallels GP and HL, the square of DH is to the square of DG, as the square of HL, or which is equal (64. 1 Sup.), the rectangle RHS is to the square of GP (4 and 22. 6 and 16. 5 Eu.), or, which is equal (Cor. 3. 14. 1 Sup.), as the square of TH to the square of GT; therefore DH is to DG, as TH is to TG (22. 6 Eu).

If from any point of a conick section or opposite sections, right lines parallel to two adjacent sides of a quadrangle inscribed in the section or sections, meet the opposite sides of the quadrangle, produced if necessary, the rectangles under the segments of these right lines, between the point in the section, and those opposite sides, are to each other, in the case of an ellipse or hyperbola, as the squares of the semidiameters to which they are parallel, and, in the case of a parabola, as the parameters of the diameters, whose ordinates are parallel to them.

Part 1. Let the inscribed quadrangle be a trapezium ABCD, see fig. 1 and 2, having two of the opposite sides AD and BC parallel; and from any point P in the section, let two right lines PK and PH be drawn, parallel to the adjacent sides AD and AB of the trapezium, meeting its opposite sides in the points K and L, G and H; the rectangle KPL and GPH are to each other, in the case of an ellipse or hyperbola, as the squares of the semidiameters to which PK und PH are parallel, and, in the case of a parabola, as the parameters of the diameters, whose ordinates are parallel to these right lines.

For let KPL meet the section again in O, and let QR be drawn, bisecting the parallels AD and BC, and meeting KL in $, it is a diameter of the section (Cor. 2. 32. 1 Sup.), and bisects in S, as well the right line PO terminated by the section (Cor. 1. 32. 1 Sup.), as KL, terminated by the right lines KC and LB, and parallel to the bisected right lines; therefore KP and OL are equal, and the rectangle KPL is equal to the rectangle