*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._
[Figure 26]
FIG. 26
*96. Definition of the polar line of a point.* Consider the quadrilateral
_K_, _L_, _M_, _N_ inscribed in the conic (Fig. 26). It determines the
four harmonic points _A_, _B_, _C_, _D_ which project from _N_ in to the
four harmonic points _M_, _B_, _K_, _O_. Now the tangents at _K_ and _M_
meet in _P_, a point on the line _AB_. The line _AB_ is thus determined
entirely by the point _O_. For if we draw any line through it, meeting the
conic in _K_ and _M_, and construct the harmonic conjugate _B_ of _O_ with
respect to _K_ and _M_, and also the two tangents at _K_ and _M_ which
meet in the point _P_, then _BP_ is the line in question. It thus appears
that the line _LON_ may be any line whatever through _O_; and since _D_,
_L_, _O_, _N_ are four harmonic points, we may describe the line _AB_ as
the locus of points which are harmonic conjugates of _O_ with respect to
the two points where any line through _O_ meets the curve.
*97.* Furthermore, since the tangents at _L_ and _N_ meet on this same
line, it appears as the locus of intersections of pairs of tangents drawn
at the extremities of chords through _O_.
*98.* This important line, which is completely determined by the point
_O_, is called the _polar_ of _O_ with respect to the conic; and the point
_O_ is called the _pole_ of the line with respect to the conic.
*99.* If a point _B_ is on the polar of _O_, then it is harmonically
conjugate to _O_ with respect to the two intersections _K_ and _M_ of the
line _BC_ with the conic. But for the same reason _O_ is on the polar of
_B_. We have, then, the fundamental theorem
_If one point lies on the polar of a second, then the second lies on the
polar of the first._
*100. Conjugate points and lines.* Such a pair of points are said to be
_conjugate_ with respect to the conic. Similarly, lines are said to be
_conjugate_ to each other with respect to the conic if one, and
consequently each, passes through the pole of the other.
[Figure 27]
FIG. 27
*101. Construction of the polar line of a given point.* Given a point _P_,
if it is within the conic (that is, if no tangents may be drawn from _P_
to the conic), we may construct its polar line by drawing through it any
two chords and joining the two points of intersection of the two pairs of
tangents at their extremities. If the point _P_ is outside the conic, we
may draw the two tangents and construct the chord of contact (Fig. 27).
*102. Self-polar triangle.* In Fig. 26 it is not difficult to see that
_AOC_ is a _self-polar_ triangle, that is, each vertex is the pole of the
opposite side. For _B_, _M_, _O_, _K_ are four harmonic points, and they
project to _C_ in four harmonic rays. The line _CO_, therefore, meets the
line _AMN_ in a point on the polar of _A_, being separated from _A_
harmonically by the points _M_ and _N_. Similarly, the line _CO_ meets
_KL_ in a point on the polar of _A_, and therefore _CO_ is the polar of
_A_. Similarly, _OA_ is the polar of _C_, and therefore _O_ is the pole of
_AC_.
*103. Pole and polar projectively related.* Another very important
theorem comes directly from Fig. 26.
_As a point __A__ moves along a straight line its polar with respect to a
conic revolves about a fixed point and describes a pencil projective to
the point-row described by __A__._
For, fix the points _L_ and _N_ and let the point _A_ move along the line
_AQ_; then the point-row _A_ is projective to the pencil _LK_, and since
_K_ moves along the conic, the pencil _LK_ is projective to the pencil
_NK_, which in turn is projective to the point-row _C_, which, finally, is
projective to the pencil _OC_, which is the polar of _A_.
*104. Duality.* We have, then, in the pole and polar relation a device
for setting up a one-to-one correspondence between the points and lines of
the plane—a correspondence which may be called projective, because to four
harmonic points or lines correspond always four harmonic lines or points.
To every figure made up of points and lines will correspond a figure made
up of lines and points. To a point-row of the second order, which is a
conic considered as a point-locus, corresponds a pencil of rays of the
second order, which is a conic considered as a line-locus. The name
’duality’ is used to describe this sort of correspondence. It is important
to note that the dual relation is subject to the same exceptions as the
one-to-one correspondence is, and must not be appealed to in cases where
the one-to-one correspondence breaks down. We have seen that there is in
Euclidean geometry one and only one ray in a pencil which has no point in
a point-row perspective to it for a corresponding point; namely, the line
parallel to the line of the point-row. Any theorem, therefore, that
involves explicitly the point at infinity is not to be translated into a
theorem concerning lines. Further, in the pencil the angle between two
lines has nothing to correspond to it in a point-row perspective to the
pencil. Any theorem, therefore, that mentions angles is not translatable
into another theorem by means of the law of duality. Now we have seen that
the notion of the infinitely distant point on a line involves the notion
of dividing a segment into any number of equal parts—in other words, of
_measuring_. If, therefore, we call any theorem that has to do with the
line at infinity or with the measurement of angles a _metrical_ theorem,
and any other kind a _projective_ theorem, we may put the case as follows:
_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.)