[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 pass 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