PROP. XIX. THEOR.

If, in a diameter of a parabola, any point be taken; the rectangle under the segments of the diameter, between its vertex and that point, and its vertex and the directrix, is to the difference of the squares, of the distances of the assumed point, from the focus, and from the directrix, as the square of the segment of the diameter, between its vertex and the directrix, is to the square of a right line, joining the focus, to the point, in which the diameter meets the directrix.

triangles DKM and DPF, KM is to KD, as PF to PD (4. 6 Eu.), or in a ratio of equality (Def. 8. 1 Sup.): from the centre K, at the distance KD, let a circle be described, which, because of the equality of KD and KM, passes through M ; and, because of the parallels PF and KM, the rectangle DPK is to the rectangle DFM, or, which is equal (Cor. 3. 36. 3 Eu.), the difference of the squares of KF and KD, as the square of PD is to the square of FD (2 and 20. 6 and Cor. 3. 22. 5 Eu).

PROP. XX. THEOR.

If two right lines, parallel to each other, both cut in two points, or one of them touch, and the other so cut, a parabola, and meet a diameter; the rectangles under the segments of the secants, or square of the segment of the tangent, between the diameter, and the point or points, wherein they meet the parabola, are to each other, as the segments of the diameter, between its vertex and the parallets.

Let SKT and skt be two secants, or SKT a secant, and RL a tangent to a parabola, parallel to each other; the secants cutting the parabola in S and T, s and t, and meeting a diameter DK in K and k, and the directrix DG in G and g; and the tangent touching the parabola in R, and meeting the diameter in L, and the directrix in H. The rectangles SKT and skt, or the rectangle SKT and square or RL, are to each other, as the segments KP, kp and LP.

Join DF, KF, kF and LF; and first, let the parallel right lines which meet the diameter, be a secant, as ST, and a tangent, as RL; and since the determining ratio in the parabola is the ratio of equality [Schol. 6. 1 Sup.], the right lines KD, kD and LD have to the distances of the points K, k and L from the directrix, the determining ratio; whence the rectangle SAT is to the difference of the squares of KF and KD, as the square of G is to the difference of the squares of KG and KD (13 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 (19. 1 Sup. and Theor. 3. 15. 5 Eu.), therefore, 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 compound 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 equal (4. 6 Eu.), and therefore those, of the square

of KG to the square of KD, and of the square of LH 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 of LH to the difference of the squares of LH and LD (Schol. 18. 5 Eu.), therefore the ratios compounded of these equal ratios are equal (22. 5 Eu.), namely, the ratios of the rectangle SKT to the rectangle DPK, and of the square of RL to the rectangle DPL, and by alternating, the rectangle SKT is to the square 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.

In like manner, if both the parallels be secants, as ST and st, it may be proved, that the rectangles SKT and skt have the same ratio to each other, as the segments of the diameter PK and Pk.

PROP. XXI. THEOR.

If a right line, touching a parabola, or cutting it in two points, meet two diameters; the squares of the segments of the tangent, or rectangles under the segments of the secant, between the diameters, and the point or points, wherein the tangent or secant meets the parabola, are to each other, as the segments of the diameters, between their vertices, and the points, in which they meet the tangent or secant.

Let DK and GL

be two diameters of
a parabola, meeting D
the section in Pand

Q, and the directrix
in D and G, and
let a tangent XZ,
touching the para-
bola in R, meet
these diameters in
X and Z, and the
directrix DG in I,
or a secant ST,
meet the same diam-
eters in K and L,
the section in S and
T, and the direc-
trix in H; the H
squares of RX and

G

RZ are to each other, as the segments of XP and ZQ; and the rectangles SKT, and SLT are to each other, as the segments PK and QL,

And first, the rectangle SKT and SLT are to each other, as the segments PK and QL; let F be the focus, and join FD, FP, FK and FL, let fall the perpendicular FO on DK, meeting GL in E, and on DK take OV equal to OD and join FV, through K, draw KM parallel to FP, meeting DF produced in M.

The right line FV is equal to FD (4. 1 Eu.), and PF to PD (Def. 8. 1 Sup.), therefore the triangles DFV and DPF are isosceles, and, having the angle PDF at the base of each common, are equiangular; therefore DV is to DF, as DF is to DP (4. 6 Eu.); or, which is equal, because of the parallels PF and KM (2. 6 Eu.), as FM is to PK; therefore the rectangle under DV, or twice DO 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 KF (Cor. 3. 36. 3 Eu.); therefore the rectangle under twice DO and PK is equal to the difference of the squares of KD and KF.

In like manner it may be proved, that the rectangle under twice GE or twice DO and QL, is equal to the difference of the squares of LG and LF.

But the rectangle SKT is to the difference of the squares of KD and KF, as the square of KH is to the difference of the squares of KH and KD (13, 1 Sup.), and the rectangle SLT is to the difference of the squares of LH and LF, as the square of LH is to the difference of the squares of LH and LG (by the same); and, because of the equiangular triangles KHD and LHG, the ratios of KH to KD and of LH to LG are equal (4. 6. Eu.), and therefore the ratios of the square of KH to the square of KD and of the square of LH to the square of LG (20. 6 and Cor. 3. 22. 5 Eu.), and therefore the ratios of the square of KH to the difference of the squares of KH and KD and of the square of LH to the difference of the squares of LH and LG (Schol. 18. 5 Eu.); therefore the ratios equal to them of the rectangle SKT to the difference of the squares of KD and KF, and of the rectangle SLT to the difference of the squares of LG and LF are equal; but the difference of the squares of KD and KF is above proved to be equal to the rectangle under twice DO and PK, and the difference of the squares of LG and LF, to the rectangle

under twice DO and QL; therefore the rectangle SKT is to the rectangle under twic. DO and PK, as the rectangle SLT is to the rectangle under twice DO and QL, and by alternating, the rectangle SKT is to the rectangle SLT, as the rectangle under twice DO and PK is to the rectangle under twice DO and QL (16.5 Eu.), or, the side twice DO being common to the two last terms, as PK is to QL(1. 6 Eu).

In like manner, it may be proved, in the case of the tangent XZ, that the rectangles under twice DO and the segments XP. and Z are severally equal to the difference of the squares of XD and XF and of ZG and ZF ; also that the square of RX is to the difference of the squares of XD and XF, as the square of RZ is to the difference of the squares of ZG and ZF, and, substituting for the differences of the squares of XD and XF and of ZG and ZF, what may be proved equal to them in like manner as above, the rectangles under twice DO and XP and under twice DO and Z, the square of RX is to the rectangle under twice DO and X, as the square of RZ is to the rectangle under twice DO and ZQ, and, alternating, the square of RX is to the square of RZ, as the rectangle under twice DO and XP is to the rectangle under twice DO and ZQ (16. 5 Eu.), or the side twice DO being common to the two last terms, as XP is to ZQ (1.6 Eu).

Scholium. If, in this proposition, the secant ST, meeting the diameters DK and GL in K and L, instead of meeting the directrix, were parallel to it, the truth of the proposition might be shewn, by drawing through K and L, right lines, parallel to each other, each of them cutting the section in two points and meeting the directrix; the rectangles SKT and SLT would be to each other, as the rectangles under the segments of the parallel secants, between the right line ST and the section (Cor. 4. 14. 1 Sup.), and therefore, as easily follows from this proposition, and the preceding, as the right lines PK and QL. A like observation is applicable to the case of a tangent, and to the 17th, 18th and 20th propositions of this book.

PROP. XXII. THEOR.

An axis of a conick section, bisects all right lines, terminated by the section, and ordinately applied thereto,