FIG. 17

For let the vertices be _A_, _B_, _C_, and _D_, and call the vertex _A_ the point 1, 6; _B_, the point 2; _C_, the point 3, 4; and _D_, the point 5 (Fig. 16). Pascal's theorem then indicates that _L = AB-CD_, _M = AD-BC_, and _N_, which is the intersection of the tangents at _A_ and _C_, are all on a straight line _u_. But if we were to call _A_ the point 2, _B_ the point 6, 1, _C_ the point 5, and _D_ the point 4, 3, then the intersection _P_ of the tangents at _B_ and _D_ are also on this same line _u_. Thus _L_, _M_, _N_, and _P_ are four points on a straight line. The consequences of this theorem are so numerous and important that we shall devote a separate chapter to them.

*77. Inscribed triangle.* Finally, three of the vertices of the hexagon may coalesce, giving a triangle inscribed in a conic. Pascal's theorem then reads as follows (Fig. 17) for this case:

_The three tangents at the vertices of a triangle inscribed in a conic meet the opposite sides in three points on a straight line._

[Figure 18]

FIG. 18

*78. Degenerate conic.* If we apply Pascal's theorem to a degenerate conic made up of a pair of straight lines, we get the following theorem (Fig. 18):

_If three points, __A__, __B__, __C__, are chosen on one line, and three points, __A'__, __B'__, __C'__, are chosen on another, then the three points __L = AB'-A'B__, __M = BC'-B'C__, __N = CA'-C'A__ are all on a straight line._

PROBLEMS

1. In Fig. 12, select different lines _u_ and trace the locus of the center of perspectivity _M_ of the lines _u_ and _u'_.

2. Given four points, _A_, _B_, _C_, _D_, in the plane, construct a fifth point _P_ such that the lines _PA_, _PB_, _PC_, _PD_ shall be four harmonic lines.

_Suggestion._ Draw a line _a_ through the point _A_ such that the four lines _a_, _AB_, _AC_, _AD_ are harmonic. Construct now a conic through _A_, _B_, _C_, and _D_ having _a_ for a tangent at _A_.

3. Where are all the points _P_, as determined in the preceding question, to be found?

4. Select any five points in the plane and draw the tangent to the conic through them at each of the five points.

5. Given four points on the conic, and the tangent at one of them, to construct the conic. ("To construct the conic" means here to construct as many other points as may be desired.)

6. Given three points on the conic, and the tangent at two of them, to construct the conic.

7. Given five points, two of which are at infinity in different directions, to construct the conic. (In this, and in the following examples, the student is supposed to be able to draw a line parallel to a given line.)

8. Given four points on a conic (two of which are at infinity and two in the finite part of the plane), together with the tangent at one of the finite points, to construct the conic.

9. The tangents to a curve at its infinitely distant points are called its _asymptotes_ if they pa.s.s through a finite part of the plane. Given the asymptotes and a finite point of a conic, to construct the conic.

10. Given an asymptote and three finite points on the conic, to determine the conic.

11. Given four points, one of which is at infinity, and given also that the line at infinity is a tangent line, to construct the conic.

CHAPTER V - PENCILS OF RAYS OF THE SECOND ORDER

*79. Pencil of rays of the second order defined.* If the corresponding points of two projective point-rows be joined by straight lines, a system of lines is obtained which is called a pencil of rays of the second order.

This name arises from the fact, easily shown (-- 57), that at most two lines of the system may pa.s.s through any arbitrary point in the plane. For if through any point there should pa.s.s three lines of the system, then this point might be taken as the center of two projective pencils, one projecting one point-row and the other projecting the other. Since, now, these pencils have three rays of one coincident with the corresponding rays of the other, the two are identical and the two point-rows are in perspective position, which was not supposed.

[Figure 19]

FIG. 19

*80. Tangents to a circle.* To get a clear notion of this system of lines, we may first show that the tangents to a circle form a system of this kind. For take any two tangents, _u_ and _u'_, to a circle, and let _A_ and _B_ be the points of contact (Fig. 19). Let now _t_ be any third tangent with point of contact at _C_ and meeting _u_ and _u'_ in _P_ and _P'_ respectively. Join _A_, _B_, _P_, _P'_, and _C_ to _O_, the center of the circle. Tangents from any point to a circle are equal, and therefore the triangles _POA_ and _POC_ are equal, as also are the triangles _P'OB_ and _P'OC_. Therefore the angle _POP'_ is constant, being equal to half the constant angle _AOC + COB_. This being true, if we take any four harmonic points, _P__1_, _P__2_, _P__3_, _P__4_, on the line _u_, they will project to _O_ in four harmonic lines, and the tangents to the circle from these four points will meet _u'_ in four harmonic points, _P'__1_, _P'__2_, _P'__3_, _P'__4_, because the lines from these points to _O_ inclose the same angles as the lines from the points _P__1_, _P__2_, _P__3_, _P__4_ on _u_. The point-row on _u_ is therefore projective to the point-row on _u'_. Thus the tangents to a circle are seen to join corresponding points on two projective point-rows, and so, according to the definition, form a pencil of rays of the second order.

*81. Tangents to a conic.* If now this figure be projected to a point outside the plane of the circle, and any section of the resulting cone be made by a plane, we can easily see that the system of rays tangent to any conic section is a pencil of rays of the second order. The converse is also true, as we shall see later, and a pencil of rays of the second order is also a set of lines tangent to a conic section.

*82.* The point-rows _u_ and _u'_ are, themselves, lines of the system, for to the common point of the two point-rows, considered as a point of _u_, must correspond some point of _u'_, and the line joining these two corresponding points is clearly _u'_ itself. Similarly for the line _u_.

*83. Determination of the pencil.* We now show that _it is possible to a.s.sign arbitrarily three lines, __a__, __b__, and __c__, of __ the system (besides the lines __u__ and __u'__); but if these three lines are chosen, the system is completely determined._

This statement is equivalent to the following:

_Given three pairs of corresponding points in two projective point-rows, it is possible to find a point in one which corresponds to any point of the other._

We proceed, then, to the solution of the fundamental

PROBLEM. _Given three pairs of points, __AA'__, __BB'__, and __CC'__, of two projective point-rows __u__ and __u'__, to find the point __D'__ of __u'__ which corresponds to any given point __D__ of __u__._

[Figure 20]

FIG. 20