[Figure 32]
FIG. 32
*121. Fundamental theorem.* The important theorem concerning two complete
quadrangles (§ 26), upon which the theory of four harmonic points was
based, can easily be extended to the case where the four lines _KL_,
_K’L’_, _MN_, _M’N’_ do not all meet in the same point _A_, and the more
general theorem that results may also be made the basis of a theory no
less important, which has to do with six points on a line. The theorem is
as follows:
_Given two complete quadrangles, __K__, __L__, __M__, __N__ and __K’__,
__L’__, __M’__, __N’__, so related that __KL__ and __K’L’__ meet in __A__,
__MN__ and __M’N’__ in __A’__, __KN__ and __K’N’__ in __B__, __LM__ and
__L’M’__ in __B’__, __LN__ and __L’N’__ in __C__, and __KM__ and __K’M’__
in __C’__, then, if __A__, __A’__, __B__, __B’__, and __C__ are in a
straight line, the point __C’__ also lies on that straight line._
The theorem follows from Desargues’s theorem (Fig. 32). It is seen that
_KK’_, _LL’_, _MM’_, _NN’_ all meet in a point, and thus, from the same
theorem, applied to the triangles _KLM_ and _K’L’M’_, the point _C’_ is on
the same line with _A_ and _B’_. As in the simpler case, it is seen that
there is an indefinite number of quadrangles which may be drawn, two sides
of which go through _A_ and _A’_, two through _B_ and _B’_, and one
through _C_. The sixth side must then go through _C’_. Therefore,
*122.* _Two pairs of points, __A__, __A’__ and __B__, __B’__, being
given, then the point __C’__ corresponding to any given point __C__ is
uniquely determined._
The construction of this sixth point is easily accomplished. Draw through
_A_ and _A’_ any two lines, and cut across them by any line through _C_ in
the points _L_ and _N_. Join _N_ to _B_ and _L_ to _B’_, thus determining
the points _K_ and _M_ on the two lines through _A_ and _A’_, The line
_KM_ determines the desired point _C’_. Manifestly, starting from _C’_, we
come in this way always to the same point _C_. The particular quadrangle
employed is of no consequence. Moreover, since one pair of opposite sides
in a complete quadrangle is not distinguishable in any way from any other,
the same set of six points will be obtained by starting from the pairs
_AA’_ and _CC’_, or from the pairs _BB’_ and _CC’_.
*123. Definition of involution of points on a line.*
_Three pairs of points on a line are said to be in involution if through
each pair may be drawn a pair of opposite sides of a complete quadrangle.
If two pairs are fixed and one of the third pair describes the line, then
the other also describes the line, and the points of the line are said to
be paired in the involution determined by the two fixed pairs._
[Figure 33]
FIG. 33
*124. Double-points in an involution.* The points _C_ and _C’_ describe
projective point-rows, as may be seen by fixing the points _L_ and _M_.
The self-corresponding points, of which there are two or none, are called
the _double-points_ in the involution. It is not difficult to see that the
double-points in the involution are harmonic conjugates with respect to
corresponding points in the involution. For, fixing as before the points
_L_ and _M_, let the intersection of the lines _CL_ and _C’M_ be _P_ (Fig.
33). The locus of _P_ is a conic which goes through the double-points,
because the point-rows _C_ and _C’_ are projective, and therefore so are
the pencils _LC_ and _MC’_ which generate the locus of _P_. Also, when _C_
and _C’_ fall together, the point _P_ coincides with them. Further, the
tangents at _L_ and _M_ to this conic described by _P_ are the lines _LB_
and _MB_. For in the pencil at _L_ the ray _LM_ common to the two pencils
which generate the conic is the ray _LB’_ and corresponds to the ray _MB_
of _M_, which is therefore the tangent line to the conic at _M_. Similarly
for the tangent _LB_ at _L_. _LM_ is therefore the polar of _B_ with
respect to this conic, and _B_ and _B’_ are therefore harmonic conjugates
with respect to the double-points. The same discussion applies to any
other pair of corresponding points in the involution.
[Figure 34]
FIG. 34
*125. Desargues’s theorem concerning conics through four points.* Let
_DD’_ be any pair of points in the involution determined as above, and
consider the conic passing through the five points _K_, _L_, _M_, _N_,
_D_. We shall use Pascal’s theorem to show that this conic also passes
through _D’_. The point _D’_ is determined as follows: Fix _L_ and _M_ as
before (Fig. 34) and join _D_ to _L_, giving on _MN_ the point _N’_. Join
_N’_ to _B_, giving on _LK_ the point _K’_. Then _MK’_ determines the
point _D’_ on the line _AA’_, given by the complete quadrangle _K’_, _L_,
_M_, _N’_. Consider the following six points, numbering them in order: _D
= 1_, _D’ = 2_, _M = 3_, _N = 4_, _K = 5_, and _L = 6_. We have the
following intersections: _B = (12-45)_, _K’ = (23-56)_, _N’ = (34-61)_;
and since by construction _B_, _N_, and _K’_ are on a straight line, it
follows from the converse of Pascal’s theorem, which is easily
established, that the six points are on a conic. We have, then, the
beautiful theorem due to Desargues:
_The system of conics through four points meets any line in the plane in
pairs of points in involution._
*126.* It appears also that the six points in involution determined by
the quadrangle through the four fixed points belong also to the same
involution with the points cut out by the system of conics, as indeed we
might infer from the fact that the three pairs of opposite sides of the
quadrangle may be considered as degenerate conics of the system.
*127. Conics through four points touching a given line.* It is further
evident that the involution determined on a line by the system of conics
will have a double-point where a conic of the system is tangent to the
line. We may therefore infer the theorem
_Through four fixed points in the plane two conics or none may be drawn
tangent to any given line._
[Figure 35]
FIG. 35
*128. Double correspondence.* We have seen that corresponding points in
an involution form two projective point-rows superposed on the same
straight line. Two projective point-rows superposed on the same straight
line are, however, not necessarily in involution, as a simple example will
show. Take two lines, _a_ and _a’_, which both revolve about a fixed point
_S_ and which always make the same angle with each other (Fig. 35). These
lines cut out on any line in the plane which does not pass through _S_ two
projective point-rows, which are not, however, in involution unless the
angle between the lines is a right angles. For a point _P_ may correspond
to a point _P’_, which in turn will correspond to some other point than
_P_. The peculiarity of point-rows in involution is that any point will
correspond to the same point, in whichever point-row it is considered as
belonging. In this case, if a point _P_ corresponds to a point _P’_, then
the point _P’_ corresponds back again to the point _P_. The points _P_ and
_P’_ are then said to _correspond doubly_. This notion is worthy of
further study.
[Figure 36]
FIG. 36
*129. Steiner’s construction.* It will be observed that the solution of
the fundamental problem given in § 83, _Given three pairs of points of two
protective point-rows, to construct other pairs_, cannot be carried out if
the two point-rows lie on the same straight line. Of course the method may
be easily altered to cover that case also, but it is worth while to give
another solution of the problem, due to Steiner, which will also give
further information regarding the theory of involution, and which may,
indeed, be used as a foundation for that theory. Let the two point-rows
_A_, _B_, _C_, _D_, ... and _A’_, _B’_, _C’_, _D’_, ... be superposed on
the line _u_. Project them both to a point _S_ and pass any conic _κ_
through _S_. We thus obtain two projective pencils, _a_, _b_, _c_, _d_,
... and _a’_, _b’_, _c’_, _d’_, ... at _S_, which meet the conic in the
points _α_, _β_, _γ_, _δ_, ... and _α’_, _β’_, _γ’_, _δ’_, ... (Fig. 36).
Take now _γ_ as the center of a pencil projecting the points _α’_, _β’_,
_δ’_, ..., and take _γ’_ as the center of a pencil projecting the points
_α_, _β_, _δ_, .... These two pencils are projective to each other, and
since they have a self-correspondin ray in common, they are in perspective
position and corresponding rays meet on the line joining _(γα’, γ’α)_ to
_(γβ’, γ’β)_. The correspondence between points in the two point-rows on
_u_ is now easily traced.
*130. Application of Steiner’s construction to double correspondence.*
Steiner’s construction throws into our hands an important theorem
concerning double correspondence: _If two projective point-rows,
superposed on the same line, have one pair of points which correspond to
each other doubly, then all pairs correspond to each other doubly, and the
line is paired in involution._ To make this appear, let us call the point
_A_ on _u_ by two names, _A_ and _P’_, according as it is thought of as
belonging to the one or to the other of the two point-rows. If this point
is one of a pair which correspond to each other doubly, then the points
_A’_ and _P_ must coincide (Fig. 37). Take now any point _C_, which we
will also call _R’_. We must show that the corresponding point _C’_ must
also coincide with the point _B_. Join all the points to _S_, as before,
and it appears that the points α and _π’_ coincide, as also do the points
_α’π_ and _γρ’_. By the above construction the line _γ’ρ_ must meet _γρ’_
on the line joining _(γα’, γ’α)_ with _(γπ’, γ’π)_. But these four points
form a quadrangle inscribed in the conic, and we know by § 95 that the
tangents at the opposite vertices _γ_ and _γ’_ meet on the line _v_. The
line _γ’ρ_ is thus a tangent to the conic, and _C’_ and _R_ are the same
point. That two projective point-rows superposed on the same line are also
in involution when one pair, and therefore all pairs, correspond doubly
may be shown by taking _S_ at one vertex of a complete quadrangle which
has two pairs of opposite sides going through two pairs of points. The
details we leave to the student.
[Figure 37]
FIG. 37
[Figure 38]
FIG. 38
*131. Involution of points on a point-row of the second order.* It is
important to note also, in Steiner’s construction, that we have obtained
two point-rows of the second order superposed on the same conic, and have
paired the points of one with the points of the other in such a way that
the correspondence is double. We may then extend the notion of involution
to point-rows of the second order and say that _the points of a conic are
paired in involution when they are corresponding __ points of two
projective point-rows superposed on the conic, and when they correspond to
each other doubly._ With this definition we may prove the theorem: _The
lines joining corresponding points of a point-row of the second order in
involution all pass through a fixed point __U__, and the line joining any
two points __A__, __B__ meets the line joining the two corresponding
points __A’__, __B’__ in the points of a line __u__, which is the polar of
__U__ with respect to the conic._ For take _A_ and _A’_ as the centers of
two pencils, the first perspective to the point-row _A’_, _B’_, _C’_ and
the second perspective to the point-row _A_, _B_, _C_. Then, since the
common ray of the two pencils corresponds to itself, they are in
perspective position, and their axis of perspectivity _u_ (Fig. 38) is the
line which joins the point _(AB’, A’B)_ to the point _(AC’, A’C)_. It is
then immediately clear, from the theory of poles and polars, that _BB’_
and _CC’_ pass through the pole _U_ of the line _u_.
*132. Involution of rays.* The whole theory thus far developed may be
dualized, and a theory of lines in involution may be built up, starting
with the complete quadrilateral. Thus,
_The three pairs of rays which may be drawn from a point through the three
pairs of opposite vertices of a complete quadrilateral are said to be in
involution. If the pairs __aa’__ and __bb’__ are fixed, and the line __c__
describes a pencil, the corresponding line __c’__ also describes a pencil,
and the rays of the pencil are said to be paired in the involution
determined by __aa’__ and __bb’__._
*133. Double rays.* The self-corresponding rays, of which there are two
or none, are called _double rays_ of the involution. Corresponding rays of
the involution are harmonic conjugates with respect to the double rays. To
the theorem of Desargues (§ 125) which has to do with the system of conics
through four points we have the dual:
_The tangents from a fixed point to a system of conics tangent to four
fixed lines form a pencil of rays in involution._
*134.* If a conic of the system should go through the fixed point, it is
clear that the two tangents would coincide and indicate a double ray of
the involution. The theorem, therefore, follows:
_Two conics or none may be drawn through a fixed point to be tangent to
four fixed lines._
*135. Double correspondence.* It further appears that two projective
pencils of rays which have the same center are in involution if two pairs
of rays correspond to each other doubly. From this it is clear that we
might have deemed six rays in involution as six rays which pass through a
point and also through six points in involution. While this would have
been entirely in accord with the treatment which was given the
corresponding problem in the theory of harmonic points and lines, it is
more satisfactory, from an aesthetic point of view, to build the theory of
lines in involution on its own base. The student can show, by methods
entirely analogous to those used in the second chapter, that involution is
a projective property; that is, six rays in involution are cut by any
transversal in six points in involution.
*136. Pencils of rays of the second order in involution.* We may also
extend the notion of involution to pencils of rays of the second order.
Thus, _the tangents to a conic are in involution when they are
corresponding rays of two protective pencils of the second order
superposed upon the same conic, and when they correspond to each other
doubly._ We have then the theorem:
*137.* _The intersections of corresponding rays of a pencil of the second
order in involution are all on a straight line __u__, and the intersection
of any two tangents __ab__, when joined to the intersection of the
corresponding tangents __a’b’__, gives a line which passes through a fixed
point __U__, the pole of the line __u__ with respect to the conic._
*138. Involution of rays determined by a conic.* We have seen in the
theory of poles and polars (§ 103) that if a point _P_ moves along a line
_m_, then the polar of _P_ revolves about a point. This pencil cuts out on
_m_ another point-row _P’_, projective also to _P_. Since the polar of _P_
passes through _P’_, the polar of _P’_ also passes through _P_, so that
the correspondence between _P_ and _P’_ is double. The two point-rows are
therefore in involution, and the double points, if any exist, are the
points where the line _m_ meets the conic. A similar involution of rays
may be found at any point in the plane, corresponding rays passing each
through the pole of the other. We have called such points and rays
_conjugate_ with respect to the conic (§ 100). We may then state the
following important theorem:
*139.* _A conic determines on every line in its plane an involution of
points, corresponding points in the involution __ being conjugate with
respect to the conic. The double points, if any exist, are the points
where the line meets the conic._
*140.* The dual theorem reads: _A conic determines at every point in the
plane an involution of rays, corresponding rays being conjugate with
respect to the conic. The double rays, if any exist, are the tangents from
the point to the conic._
PROBLEMS
1. Two lines are drawn through a point on a conic so as always to make
right angles with each other. Show that the lines joining the points where
they meet the conic again all pass through a fixed point.
2. Two lines are drawn through a fixed point on a conic so as always to
make equal angles with the tangent at that point. Show that the lines
joining the two points where the lines meet the conic again all pass
through a fixed point.
3. Four lines divide the plane into a certain number of regions.
Determine for each region whether two conics or none may be drawn to pass
through points of it and also to be tangent to the four lines.
4. If a variable quadrangle move in such a way as always to remain
inscribed in a fixed conic, while three of its sides turn each around one
of three fixed collinear points, then the fourth will also turn around a
fourth fixed point collinear with the other three.
5. State and prove the dual of problem 4.
6. Extend problem 4 as follows: If a variable polygon of an even number
of sides move in such a way as always to remain inscribed in a fixed
conic, while all its sides but one pass through as many fixed collinear
points, then the last side will also pass through a fixed point collinear
with the others.
7. If a triangle _QRS_ be inscribed in a conic, and if a transversal _s_
meet two of its sides in _A_ and _A’_, the third side and the tangent at
the opposite vertex in _B_ and _B’_, and the conic itself in _C_ and _C’_,
then _AA’_, _BB’_, _CC’_ are three pairs of points in an involution.
8. Use the last exercise to solve the problem: Given five points, _Q_,
_R_, _S_, _C_, _C’_, on a conic, to draw the tangent at any one of them.
9. State and prove the dual of problem 7 and use it to prove the dual of
problem 8.
10. If a transversal cut two tangents to a conic in _B_ and _B’_, their
chord of contact in _A_, and the conic itself in _P_ and _P’_, then the
point _A_ is a double point of the involution determined by _BB’_ and
_PP’_.
11. State and prove the dual of problem 10.
12. If a variable conic pass through two given points, _P_ and _P’_, and
if it be tangent to two given lines, the chord of contact of these two
tangents will always pass through a fixed point on _PP’_.
13. Use the last theorem to solve the problem: Given four points, _P_,
_P’_, _Q_, _S_, on a conic, and the tangent at one of them, _Q_, to draw
the tangent at any one of the other points, _S_.
14. Apply the theorem of problem 9 to the case of a hyperbola where the
two tangents are the asymptotes. Show in this way that if a hyperbola and
its asymptotes be cut by a transversal, the segments intercepted by the
curve and by the asymptotes respectively have the same middle point.
15. In a triangle circumscribed about a conic, any side is divided
harmonically by its point of contact and the point where it meets the
chord joining the points of contact of the other two sides.