Any projective theorem involves another theorem, dual to it, obtainable by interchanging everywhere the words ‘point’ and ’line.’

105.  Self-dual theorems. The theorems of this chapter will be found, upon examination, to be self-dual; that is, no new theorem results from applying the process indicated in the preceding paragraph.  It is therefore useless to look for new results from the theorem on the circumscribed quadrilateral derived from Brianchon’s, which is itself clearly the dual of Pascal’s theorem, and in fact was first discovered by dualization of Pascal’s.

106. It should not be inferred from the above discussion that one-to-one correspondences may not be devised that will control certain of the so-called metrical relations.  A very important one may be easily found that leaves angles unaltered.  The relation called similarity leaves ratios between corresponding segments unaltered.  The above statements apply only to the particular one-to-one correspondence considered.

PROBLEMS

1.  Given a quadrilateral, construct the quadrangle polar to it with respect to a given conic.

2.  A point moves along a straight line.  Show that its polar lines with respect to two given conics generate a point-row of the second order.

3.  Given five points, draw the polar of a point with respect to the conic passing through them, without drawing the conic itself.

4.  Given five lines, draw the polar of a point with respect to the conic tangent to them, without drawing the conic itself.

5.  Dualize problems 3 and 4.

6.  Given four points on the conic, and the tangent at one of them, draw the polar of a given point without drawing the conic.  Dualize.

7.  A point moves on a conic.  Show that its polar line with respect to another conic describes a pencil of rays of the second order.

Suggestion. Replace the given conic by a pair of protective pencils.

8.  Show that the poles of the tangents of one conic with respect to another lie on a conic.

9.  The polar of a point A with respect to one conic is a, and the pole of a with respect to another conic is A’.  Show that as A travels along a line, A’ also travels along another line.  In general, if A describes a curve of degree n, show that A’ describes another curve of the same degree n. (The degree of a curve is the greatest number of points that it may have in common with any line in the plane.)

CHAPTER VII — METRICAL PROPERTIES OF THE CONIC SECTIONS

107.  Diameters.  Center. After what has been said in the last chapter one would naturally expect to get at the metrical properties of the conic sections by the introduction of the infinite elements in the plane.  Entering into the theory of poles and polars with these elements, we have the following definitions: