Cor. 1. From this proposition and scholium, it appears, how, the focus and principal vertex of a parabola, or the focuses of an ellipse or hyperbola being given, a right line may be drawn touching the section in a given point. Namely, by drawing from the given point in the case of a parabola, two right lines, one to the focus, and the other perpendicularly to the directrix} and, in the other cases, right lines to the focuses; the right line, bisecting the angle, formed by the right lines so drawn, in the cases of a hyperbola or parabola, and the angle in continuation thereto, in the case of an ellipse, is a tangent to the section (10.. 1 Sup). But if the given point be the vertex of an axis, a right line, drawn through the given point, perpendicular to the axis, is the tangent required (Schol. 10. 1 Sup).

And since these focuses and principal vertices, are necessary parts of these figures, as is manifest from the definition of them; it follows, that a right line may touch a conick section in any point thereof.

Cor. 2. It also appears from t'ais proposition, that, as a perpendicular is the nearest distance of a point from a right line (Cor. 1. 19. 1 Eu.); so the sum of two right lines (EP and PF, see fig. 1 of this prop.), drawn from two points (E and F) on the same side of a right line (GK), to any point in it (as P), is least, when the right line (GK), to which they are drawn, makes equal angles (EPG and FPK) with them, towards opposite parts (G and K).

For it is demonstrated in this proposition, that the sum of any other two, as EG and GF, so drawn, is greater than the sum of thesc.

Cor. 3. And hence appears a method of drawing to a given right line (GK), from two points (E and F) on the same side of it, right lines, to make with it equal angles, at the same point, towards opposite parts. Namely, by letting fall the perpendicular FK from one of the given points F, on GK, taking, on FK produced, KH equal to FK, and joining EH, which would cut GK in the point P required.

For the angle FPK is equal to HPK (4. 1 Eu.), or its equal (15. 1 Eu.) EPG.

Cor. 4. The differences of the distances, of two right lines (EP and PF see fig. 2), drawn from two points (E and F), on different sides of and at unequal distances from, a right line (GR), to any point (P) in it, is greatest, when the right line (GK), to which they are drawn, makes equal angles (EPK and FPK) with them, towards the same part.

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For it is demonstrated in this proposition, that the difference of any other two, as EG and GF, so drawn, is less than the difference of these.

Cor. 5. And hence appears a method of drawing to a given right line (GK), from two points (E and F), on different sides of, and at unequal distances from it, right lines, to make with it equal angles, at the same point, and towards the same part. Namely, by letting fall the perpendicular FK, from one of the given points F, on GK; taking on FK produced, KH equal to FK, and joining EH ; which, because the points E and F, or E and H, are at unequal distances from GK, being produced towards the least distance HK, would meet GK, let it, so produced, meet it in P, and join FP.

The angles HPK or EPK and FPK are equal (4. 1 Eu).

Cor. 6. No right line can be drawn, in any conick section (see fig. 1, 2 and 3), between the right line GPK, bisecting the angle FPH, and the section, and of course any right line, drawn through P, which makes unequal angles with the right lines PF and PH, enters the section on the part of the point P, whether towards G or K, on which the angle formed with PF is less than that formed with PH.

For such a right line would divide the angle FPH unequally; whence, in the case of the ellipse and hyperbola, see fig. 1 and 2, right lines being drawn from E and F, to make equal angles with that right line at the same point, towards opposite parts in fig. 1, and to the same part in fig. 2 (by cor. 3 and 5 above), the sum of these, in the former case, would be less than of EP and PF (Cor. 2 above), and the difference in the latter case, greater than of EP and PF (cor. 4 above); therefore, in both cases, the point, to which the right lines are so drawn from E and F, is within the section (4. 1 Sup.); therefore the right line, so dividing the angle FPH unequally, enters the section; and, that it enters it on that side of P, on which the angle it makes with PF is less, is manifest, since, on the other side of P, it falls without GPK with respect to the section, and of course wholly without the section.

And, in the case of a parabola, see adjacent fig. a right line PM, which makes a less angle with PF than with PH, enters the section on the side of P which is towards F; for, from the greater angle MPH take MPN equal to MPF, make PN equal to PH, join NH, through N draw QNR at right angles to the directrix QH, meeting PM in R, and join RF.

Because PN is equal to

H

n

N

M

PH, the triangle PHN is isosceles, therefore the angle PHN, being equal to PNH (5. 1 Eu.), is less than the right angle PHQ (32. 1 Eu.), therefore the point N falls between Q and R, and because RP, PF and the included angle RPF, are severally equal to RP, PN and the included angle RPN, RF is equal to RN (4. 1 Eu.), and therefore less than QR, and so the point R is within the section (4. 1 Sup).

And if a right line Pm make, on the other side of P, a less angle with PF than with PH; by using a similar construction and demonstration, as in the preceding case, only substituting, the small letters m, n, q, r for the corresponding capitals, it may be shewn, that the point r in Pm is within the section; so that, in the case of a parabola, as well as of the other sections, a right line passing through P, and making unequal angles with PF and PH, enters the section on the part of P, on which the angle formed with PF, is less than that formed with PH.

Cor. 7. Through any point of a conick section, there can be drawn but one right line, touching it in that point. It having been proved in the preceding corollary, that any other right line, except that bisecting the angle FPH, enters the section, and is not of course a tangent (Def. 10 of this).

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The condition in the second corollary to this proposition, towards opposite parts, is quite necessary; for if the points E and F, see fig. 1, be at unequal distances from GK, the right line EF, being produced toward the least distance, would meet GK, and form the same, and of course, equal angles with GK, but to the same part; a like observation is applicable to the condition in the fourth corollary, towards the same part; for the points E and F, see fig. 2, being on different sides of GK, the right line EF would intersect GK produced, and form with it equal angles, but towards opposite parts (15. 1 Eu).

PROP. XI. THEOR.

A tangent (GPK, see fig. 1, 2 and 3 of prec. prop.) to a conick section, bisects, in ellipses, the angle FPH, see fig. 1), in continuation to that (EPF), formed by right lines drawn from the contact (P) to the focuses; in hyperbolas, that (EPF, see fig. 2), formed by right lines so drawn to the focuses; and, in parabolas, that (FPH, see fig. 3), formed by two right lines drawn from the contact (P), one (PF) to the focus, and the other (PH), perpendicularly to the directrix (DH).

For in every case, if the tangent PK bisects not the angle FPH, let there be drawn a right line bisecting this angle (9.1 Eu.), the right line so drawn also touches the section in P (10 1. Sup.); which cannot be (Cor. 7. 10. 1 Sup.), therefore the tangent GPK bisects in every case the angle FPH.

Cor. 1. A right line, touching a conick section in the vertex of an axis, is perpendicular to the axis; for if not, a perpendicular to the axis being drawn at the vertex, would be a tangent to the section (Schol. 10. 1 Sup.), and so two right lines would touch the section in the same point, contrary to cor. 7 10. 1 Sup.

Cor. 2. A right line, drawn from any point of a conick section, perpendicularly to the axis, is ordinately applied to it; being parallel to the tangent, passing through a vertex of the axis, which is perpendicular to the axis (prec. cor.); the part between the section and axis, being an ordinate to the axis (Def. 12. 1 Sup).

Cor. 3. An ordinate to an axis, is perpendicular to it, because the tangent of the vertex, to which the ordinate is parallel (Def. 12. 1 Sup.), is perpendicular to the axis (cor. 1 above).

Cor. 4. If a right line, joining the vertices of two diameters of a parabola, be ordinately applied to the axis, their parameters are equal.

For the segments of these diameters, between their vertices and the directrix, are equal, being opposite sides of a parallelogram, and therefore the parameters, being fourfold of these segments (Def. 16. 1 Sup).

PROP. XII. THEOR.

A right line, passing through a focus of a conick section, and any point in the adjacent directrix, is perpendicular to a right line, joining that focus, to the contact, of a tangent to the section, drawn from the same point in the directrix; and makes equal angles, with two right lines, joining that focus, to the intersections, with the section or opposite sections, of any right line, drawn from the same point in the directrix, and cutting in two points the section or sections; the equal angles being on the same or different sides, of the right line joining the directrix and focus, and towards opposite parts or the same, according as the intersections of the secant, are in the same or opposite sections.