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course the section, in the point Q (Cor. 16. 3 Eu. and Def. 21. 1 Sup).

The proof is similar, when the terminated right line is ordinately applied to the second axis of an ellipse or hyperbola.

PROP. LXXIV. THEOR.

If two circles (QP and RP, see figures to preceding prop.), touch an ellipse or hyperbola in the same point (P), and again touch the same ellipse or opposite hyperbolas in two other points (Qand R); the rectangle contained under the diameters of the circles, is equal to the square of the diameter of the section, which is conjugate to the diameter, passing through the contact of the circles. And if a circle touch a parabola in two points, the square of its diameter, is equal to the rectangle contained under the principal parameter, and the parameter of the diameter of the section passing through the contact.

Part 1. Let AB and MN be the axes of the ellipse or hyperbolas, and C the centre, the right lines PQ and PR joining the contacts are perpendicular to the axes (73. 1 and Cor, 3. 11. 1 Sup.); through the point P in which both circles touch the section, draw the common tangent (48. 1 Sup.), meeting the axes in K and L, and from P, draw PS perpendicular to KP, meeting the axes in O and S, the points O and S are the centres of the circles (1 and 19. 3 Eu.), and, because the diameters of the section passing through the contacts are equal (Cor. 3. 11. 1 and 31. 1 Sup. and 4. 1 Eu.), their conjugates are equal (Cor. 1. 53. 1 Sup. and 14. 6 Eu.]; and the triangles LPS and OPK, being right angled at P, and having the angles at S and K equal, being each the complement of the angle SLK to a right angle, are equiangular, therefore SP is to PL, as PK is to PO (4. 6 Eu.), and therefore the rectangle SPO under the semidiameters of the circles, is equal to the rectangle LPK (16. 6 Eu.), or which is equal (47. 1 Sup.), the square of the semidiameter parallel to LP, which semidiameter is conjugate to that passing through P (Def. 14. 1 Sup.), therefore the rectangle under the diameters of the circles, is equal to the square of the diameter of the section, conjugate to the diameter passing through P (23. 6 and Ax. 1. 5 Eu).

Part 2. Let PBQ in fig. 2 be a parabola, and let the circle PQ touch it in P and Q, PQ being joined is perpendicular to the axis (73. 1 and Cor. 3. 11. 1 Sup.), through P draw the common tangent (48. 1 Sup.), meeting the axis in K, and from P, draw PO perpendicular to the tangent, meeting the axis in O, the point O is the centre of the circle PQ (1 and 19. S Eu.); and the triangles OP and OHP, being right angled at P and H. and having the angle KOP common, are equiangular, therefore KO is to OP, as OP is to OH (4. 6 Eu.), and therefore the square of OP is equal to the rectangle HOK (17. 6 Eu.); but, because the rectangle under BH and the principal parameter, or, KH being double to BH (44. 1 Sup.), the rectangle under KH and half that parameter, is equal to the square of PH (23. 1 Sup.), or, which is equal (Cor. 1. 8. 6 and 17. 6 Eu.), the rectangle KHO, therefore HO is equal to half the principal parameter; and, because the rectangle under KB and the parameter of the diameter passing through P, or, KH being double to KB (44. 1 Sup.), the rectangle under KH and half the parameter of the diameter passing through P is equal to the square of PK (41. t Sup.), or, which is equal (Cor. 1. 8. 6 and 17. 6 Eu.), the rectangle OKH, therefore KO is equal to half the parameter of the diameter passing through P; whence, the square of the semidiameter OP of the circle having been just proved equal to the rectangle HOK, the square of that semidiameter is equal to the rectangle under half the principal parameter, and half the parameter of the diameter of the section passing through P, and therefore the square of the diameter of the circle, is equal to the rectangle under the principal parame ter, and the parameter of the diameter of the section passing through P (23. 6 and Ax. 1. 5 Eu).

PROP. LXXV. THEOR.

If from the vertex of the transverse axis of an ellipse or hyperbola, or the axis of a parabola, there be put in the axis towards the interior of the section, a right line equal to its parameter; a circle described about this, as a diameter, falls entirely within the section; and if from the vertex of the less axis of an ellipse, there be put in the axis, towards the interior of the section, a right line equal to its parameter; a circle described about this as a diameter, falls entirely without the section.

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Let AB in fig. 1 and 3, be the transverse axis of an ellipse or hyperbola, or in fig. 2, the second axis of an ellipse, and BH in fig. 4, the axis of a parabola, B being in every case its vertex; let there be taken on each of these axes, from B, towards the interior of the section, a right line BD, equal to its parameter; a circle described about BD as a diameter, falls, in the case of fig. 1, 3 and 4, entirely within, and, in that of fig. 2, entirely without the section.

Draw BG perpendicular to the axis, and equal to BD, and join DG; and, in the ellipse and hyperbola, let AQG be drawn, from the vertex A of the axis AB, but, in the parabola, GQ parallel to the axis; through any point E in the circle, draw EH parallel to BG, meeting the section in L, and BD, DG and GQ in H, K and Q.

Because BD and BG are equal, DH and HK are equal [4. 6 Eu.], and because of the circle, the square of EH is equal to the rectangle BHD [3 and 35. s Eu.], or BHK; but, because of the section, the square of HL is equal to the rectangle BHQ, for, in the case of fig. 1, 2 and 3, the rectangle AHB is to the square of HL, as AB is to its parameter BG [Cor. 4. 40. 1 Sup.]

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or [4. 6 Eu.], as AH to HQ, or [1. 6 Eu.], as the rectangle AHB is to the rectangle BHQ, and therefore the square of IL and rectangle BHQ, to which the rectangle AHB has the same ratio, are equal [9. 5 Eu.]; and, in the case of fig. 4, the square of HL is equal to the rectangle under BH and BG or HQ [23. 1 Sup.]; but HK is less than HQ, unless when B is a vertex of the second axis of an ellipse, in which case HK is greater than HQ; therefore, in the cases of fig. 1, 3 and 4, the rectangle · BHK is less than BHQ, and of course the square of EH less than the square of HL, and the right line EH less than HL, and therefore the circle BED is entirely within the section. In like manner it may be shewn, in the case of the second axis of an ellipse, fig. 2, that the circle is entirely without the section.

Cor. 1. Hence, if from a vertex [B] of an axis [BH] of a conick section, there be put in the axis, towards the interior of the section, a right line [BO] in the case of a principal axís, not greater, and of a less axis of an ellipse, not less than half the parameter [BD] of the same axis; that right line [BO is, in the former case, the least, and, in the latter the greatest, which can be drawn from the extreme [O] remote from the vertex, to the section.

For, in the former case, because the circle described from the centre O does not fall without the circle BED, it is entirely within the section; and since this circle, in the latter case, does not fall within the circle BED, it is entirely without the section; whence the thing proposed is manifest.

Cor. 2. From a point [O see next fig.], in the principal axis [BH] of a conick section, and within the section, whose distance from the nearer vertex [B] is greater than half its parameter, to draw the least right line, which can be drawn from that point to the section, on the same side of the axis.

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If the section be a parabola, let there be put in the axis, towards the vertex B, a right line OH, equal to half the parameter of the axis; and, if the section be an ellipse or hyperbola, let there be put in the axis, from the centre C, such right line CH, towards the vertex B, that CH may be to OH, as the axis AB is to its parameter [Cor. 1. 10. 6 Eu.], and, in every section, let there be drawn through H, a right line PHQ perpendicular to the axis, meeting the section in P, OP being joined is the right line required.

For through P, let a tangent to the section PK be drawn [48. 1 Sup.], meeting the axis BH in K, then, in the case of a parabola, the square of PH is equal to the rectangle under BH and the principal parameter [23. 1 Sup.], or, KH being double of BH [44. 1 Sup.], and HO equal to half the parameter, to the rectangle KHO, therefore the angle KPO is a right angle [17. 6 and Cor. 2. 8. 6 Eu.], and therefore OP the least right line, which can be drawn from 0 to the section on the same side of BH [Cor. 1. 73. 1 Sup).

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