[Figure 25]

FIG. 25

93. Brianchon’s theorem may also be applied to a degenerate conic made up of two points and the lines through them.  Thus(Fig. 25),

If _a__, __b__, __c__ are three lines through a point __S__, and __a’__, __b’__, __c’__ are three lines through another point __S’__, then the lines __l = (ab’, a’b)__, __m = (bc’, b’c)__, and __n = (ca’, c’a)__ all meet in a point._

94.  Law of duality. The observant student will not have failed to note the remarkable similarity between the theorems of this chapter and those of the preceding.  He will have noted that points have replaced lines and lines have replaced points; that points on a curve have been replaced by tangents to a curve; that pencils have been replaced by point-rows, and that a conic considered as made up of a succession of points has been replaced by a conic considered as generated by a moving tangent line.  The theory upon which this wonderful law of duality is based will be developed in the next chapter.

PROBLEMS

1.  Given four lines in the plane, to construct another which shall meet them in four harmonic points.

2.  Where are all such lines found?

3.  Given any five lines in the plane, construct on each the point of contact with the conic tangent to them all.

4.  Given four lines and the point of contact on one, to construct the conic. ("To construct the conic” means here to draw as many other tangents as may be desired.)

5.  Given three lines and the point of contact on two of them, to construct the conic.

6.  Given four lines and the line at infinity, to construct the conic.

7.  Given three lines and the line at infinity, together with the point of contact at infinity, to construct the conic.

8.  Given three lines, two of which are asymptotes, to construct the conic.

9.  Given five tangents to a conic, to draw a tangent which shall be parallel to any one of them.

10.  The lines a, b, c are drawn parallel to each other.  The lines a’, b’, c’ are also drawn parallel to each other.  Show why the lines (ab’, a’b), (bc’, b’c), (ca’, c’a) meet in a point. (In problems 6 to 10 inclusive, parallel lines are to be drawn.)

CHAPTER VI — POLES AND POLARS

95.  Inscribed and circumscribed quadrilaterals. The following theorems have been noted as special cases of Pascal’s and Brianchon’s theorems: 

If a quadrilateral be inscribed in a conic, two pairs of opposite sides and the tangents at opposite vertices intersect in four points, all of which lie on a straight line.

If a quadrilateral be circumscribed about a conic, the lines joining two pairs of opposite vertices and the lines joining two opposite points of contact are four lines which meet in a point.