*Project Gutenberg*. Public domain.

*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.

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: