An Elementary Course in Synthetic Projective Geometry Part 6

*66. Lines joining four points of the locus to a fifth.* Suppose that the points _S_, _S'_, _B_, _C_, and _D_ are fixed, and that four points, _A_, _A__1_, _A__2_, and _A__3_, are taken on the locus at the intersection with it of any four harmonic rays through _B_. These four harmonic rays give four harmonic points, _L_, _L__1_ etc., on the fixed ray _SD_. These, in turn, project through the fixed point _M_ into four harmonic points, _N_, _N__1_ etc., on the fixed line _DS'_. These last four harmonic points give four harmonic rays _CA_, _CA__1_, _CA__2_, _CA__3_. Therefore the four points _A_ which project to _B_ in four harmonic rays also project to _C_ in four harmonic rays. But _C_ may be any point on the locus, and so we have the very important theorem,

_Four points which are on the locus, and which project to a fifth point of the locus in four harmonic rays, project to any point of the locus in four harmonic rays._

*67.* The theorem may also be stated thus:

_The locus of points from which, four given points are seen along four harmonic rays is a point-row of the second order through them._

*68.* A further theorem of prime importance also follows:

_Any two points on the locus may be taken as the centers of two projective pencils which will generate the locus._

*69. Pascal's theorem.* The points _A_, _B_, _C_, _D_, _S_, and _S'_ may thus be considered as chosen arbitrarily on the locus, and the following remarkable theorem follows at once.

_Given six points, 1, 2, 3, 4, 5, 6, on the point-row of the second order, if we call_

_L the intersection of 12 with 45,_

_M the intersection of 23 with 56,_

_N the intersection of 34 with 61,_

_then __L__, __M__, and __N__ are on a straight line._

[Figure 13]

FIG. 13

*70.* To get the notation to correspond to the figure, we may take (Fig.

13) _A = 1_, _B = 2_, _S' = 3_, _D = 4_, _S = 5_, and _C = 6_. If we make _A = 1_, _C=2_, _S=3_, _D = 4_, _S'=5_, and. _B = 6_, the points _L_ and _N_ are interchanged, but the line is left unchanged. It is clear that one point may be named arbitrarily and the other five named in _5! = 120_ different ways, but since, as we have seen, two different a.s.signments of names give the same line, it follows that there cannot be more than 60 different lines _LMN_ obtained in this way from a given set of six points.

As a matter of fact, the number obtained in this way is in general _60_.

The above theorem, which is of cardinal importance in the theory of the point-row of the second order, is due to Pascal and was discovered by him at the age of sixteen. It is, no doubt, the most important contribution to the theory of these loci since the days of Apollonius. If the six points be called the vertices of a hexagon inscribed in the curve, then the sides 12 and 45 may be appropriately called a pair of opposite sides. Pascal's theorem, then, may be stated as follows:

_The three pairs of opposite sides of a hexagon inscribed in a point-row of the second order meet in three points on a line._

*71. Harmonic points on a point-row of the second order.* Before proceeding to develop the consequences of this theorem, we note another result of the utmost importance for the higher developments of pure geometry, which follows from the fact that if four points on the locus project to a fifth in four harmonic rays, they will project to any point of the locus in four harmonic rays. It is natural to speak of four such points as four harmonic points on the locus, and to use this notion to define projective correspondence between point-rows of the second order, or between a point-row of the second order and any fundamental form of the first order. Thus, in particular, the point-row of the second order, s, is said to be _perspectively related_ to the pencil _S_ when every ray on _S_ goes through the point on s which corresponds to it.

*72. Determination of the locus.* It is now clear that five points, arbitrarily chosen in the plane, are sufficient to determine a point-row of the second order through them. Two of the points may be taken as centers of two projective pencils, and the three others will determine three pairs of corresponding rays of the pencils, and therefore all pairs.

If four points of the locus are given, together with the tangent at one of them, the locus is likewise completely determined. For if the point at which the tangent is given be taken as the center _S_ of one pencil, and any other of the points for _S'_, then, besides the two pairs of corresponding rays determined by the remaining two points, we have one more pair, consisting of the tangent at _S_ and the ray _SS'_. Similarly, the curve is determined by three points and the tangents at two of them.

*73. Circles and conics as point-rows of the second order.* It is not difficult to see that a circle is a point-row of the second order. Indeed, take any point _S_ on the circle and draw four harmonic rays through it.

They will cut the circle in four points, which will project to any other point of the curve in four harmonic rays; for, by the theorem concerning the angles inscribed in a circle, the angles involved in the second set of four lines are the same as those in the first set. If, moreover, we project the figure to any point in s.p.a.ce, we shall get a cone, standing on a circular base, generated by two projective axial pencils which are the projections of the pencils at _S_ and _S'_. Cut across, now, by any plane, and we get a conic section which is thus exhibited as the locus of intersection of two projective pencils. It thus appears that a conic section is a point-row of the second order. It will later appear that a point-row of the second order is a conic section. In the future, therefore, we shall refer to a point-row of the second order as a conic.

[Figure 14]

FIG. 14

*74. Conic through five points.* Pascal's theorem furnishes an elegant solution of the problem of drawing a conic through five given points. To construct a sixth point on the conic, draw through the point numbered 1 an arbitrary line (Fig. 14), and let the desired point 6 be the second point of intersection of this line with the conic. The point _L = 12-45_ is obtainable at once; also the point _N = 34-61_. But _L_ and _N_ determine Pascal's line, and the intersection of 23 with 56 must be on this line.

Intersect, then, the line _LN_ with 23 and obtain the point _M_. Join _M_ to 5 and intersect with 61 for the desired point 6.

[Figure 15]

FIG. 15

*75. Tangent to a conic.* If two points of Pascal's hexagon approach coincidence, then the line joining them approaches as a limiting position the tangent line at that point. Pascal's theorem thus affords a ready method of drawing the tangent line to a conic at a given point. If the conic is determined by the points 1, 2, 3, 4, 5 (Fig. 15), and it is desired to draw the tangent at the point 1, we may call that point 1, 6.

The points _L_ and _M_ are obtained as usual, and the intersection of 34 with _LM_ gives _N_. Join _N_ to the point 1 for the desired tangent at that point.

*76. Inscribed quadrangle.* Two pairs of vertices may coalesce, giving an inscribed quadrangle. Pascal's theorem gives for this case the very important theorem

_Two pairs of opposite sides of any quadrangle inscribed in a conic meet on a straight line, upon which line also intersect the two pairs of tangents at the opposite vertices._

[Figure 16]

FIG. 16

[Figure 17]