Lehmer, Derrick Norman - An Elementary Course in Synthetic Projective Geometry

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*71. Harmonic points on a point-row of the second order.* Before
proceeding to develop the consequences of this theorem, we note another
result of the utmost importance for the higher developments of pure
geometry, which follows from the fact that if four points on the locus
project to a fifth in four harmonic rays, they will project to any point
of the locus in four harmonic rays. It is natural to speak of four such
points as four harmonic points on the locus, and to use this notion to
define projective correspondence between point-rows of the second order,
or between a point-row of the second order and any fundamental form of the
first order. Thus, in particular, the point-row of the second order, {~GREEK SMALL LETTER SIGMA~}, is
said to be _perspectively related_ to the pencil _S_ when every ray on _S_
goes through the point on {~GREEK SMALL LETTER SIGMA~} which corresponds to it.

*72. Determination of the locus.* It is now clear that five points,
arbitrarily chosen in the plane, are sufficient to determine a point-row
of the second order through them. Two of the points may be taken as
centers of two projective pencils, and the three others will determine
three pairs of corresponding rays of the pencils, and therefore all pairs.
If four points of the locus are given, together with the tangent at one of
them, the locus is likewise completely determined. For if the point at
which the tangent is given be taken as the center _S_ of one pencil, and
any other of the points for _S'_, then, besides the two pairs of
corresponding rays determined by the remaining two points, we have one
more pair, consisting of the tangent at _S_ and the ray _SS'_. Similarly,
the curve is determined by three points and the tangents at two of them.

*73. Circles and conics as point-rows of the second order.* It is not
difficult to see that a circle is a point-row of the second order. Indeed,
take any point _S_ on the circle and draw four harmonic rays through it.
They will cut the circle in four points, which will project to any other
point of the curve in four harmonic rays; for, by the theorem concerning
the angles inscribed in a circle, the angles involved in the second set of
four lines are the same as those in the first set. If, moreover, we
project the figure to any point in space, we shall get a cone, standing on
a circular base, generated by two projective axial pencils which are the
projections of the pencils at _S_ and _S'_. Cut across, now, by any plane,
and we get a conic section which is thus exhibited as the locus of
intersection of two projective pencils. It thus appears that a conic
section is a point-row of the second order. It will later appear that a
point-row of the second order is a conic section. In the future,
therefore, we shall refer to a point-row of the second order as a conic.

[Figure 14]

FIG. 14

*74. Conic through five points.* Pascal's theorem furnishes an elegant
solution of the problem of drawing a conic through five given points. To
construct a sixth point on the conic, draw through the point numbered 1 an
arbitrary line (Fig. 14), and let the desired point 6 be the second point
of intersection of this line with the conic. The point _L = 12-45_ is
obtainable at once; also the point _N = 34-61_. But _L_ and _N_ determine
Pascal's line, and the intersection of 23 with 56 must be on this line.
Intersect, then, the line _LN_ with 23 and obtain the point _M_. Join _M_
to 5 and intersect with 61 for the desired point 6.

[Figure 15]

FIG. 15

*75. Tangent to a conic.* If two points of Pascal's hexagon approach
coincidence, then the line joining them approaches as a limiting position
the tangent line at that point. Pascal's theorem thus affords a ready
method of drawing the tangent line to a conic at a given point. If the
conic is determined by the points 1, 2, 3, 4, 5 (Fig. 15), and it is
desired to draw the tangent at the point 1, we may call that point 1, 6.
The points _L_ and _M_ are obtained as usual, and the intersection of 34
with _LM_ gives _N_. Join _N_ to the point 1 for the desired tangent at
that point.

*76. Inscribed quadrangle.* Two pairs of vertices may coalesce, giving an
inscribed quadrangle. Pascal's theorem gives for this case the very
important theorem

_Two pairs of opposite sides of any quadrangle inscribed in a conic meet
on a straight line, upon which line also intersect the two pairs of
tangents at the opposite vertices._

[Figure 16]

FIG. 16

[Figure 17]

FIG. 17

For let the vertices be _A_, _B_, _C_, and _D_, and call the vertex _A_
the point 1, 6; _B_, the point 2; _C_, the point 3, 4; and _D_, the point
5 (Fig. 16). Pascal's theorem then indicates that _L = AB-CD_, _M =
AD-BC_, and _N_, which is the intersection of the tangents at _A_ and _C_,
are all on a straight line _u_. But if we were to call _A_ the point 2,
_B_ the point 6, 1, _C_ the point 5, and _D_ the point 4, 3, then the
intersection _P_ of the tangents at _B_ and _D_ are also on this same line
_u_. Thus _L_, _M_, _N_, and _P_ are four points on a straight line. The
consequences of this theorem are so numerous and important that we shall
devote a separate chapter to them.

*77. Inscribed triangle.* Finally, three of the vertices of the hexagon
may coalesce, giving a triangle inscribed in a conic. Pascal's theorem
then reads as follows (Fig. 17) for this case:

_The three tangents at the vertices of a triangle inscribed in a conic
meet the opposite sides in three points on a straight line._

[Figure 18]

FIG. 18

*78. Degenerate conic.* If we apply Pascal's theorem to a degenerate
conic made up of a pair of straight lines, we get the following theorem
(Fig. 18):

_If three points, __A__, __B__, __C__, are chosen on one line, and three
points, __A'__, __B'__, __C'__, are chosen on another, then the three
points __L = AB'-A'B__, __M = BC'-B'C__, __N = CA'-C'A__ are all on a
straight line._

PROBLEMS

1. In Fig. 12, select different lines _u_ and trace the locus of the
center of perspectivity _M_ of the lines _u_ and _u'_.

2. Given four points, _A_, _B_, _C_, _D_, in the plane, construct a fifth
point _P_ such that the lines _PA_, _PB_, _PC_, _PD_ shall be four
harmonic lines.

_Suggestion._ Draw a line _a_ through the point _A_ such that the four
lines _a_, _AB_, _AC_, _AD_ are harmonic. Construct now a conic through
_A_, _B_, _C_, and _D_ having _a_ for a tangent at _A_.

3. Where are all the points _P_, as determined in the preceding question,
to be found?

4. Select any five points in the plane and draw the tangent to the conic
through them at each of the five points.

5. Given four points on the conic, and the tangent at one of them, to
construct the conic. ("To construct the conic" means here to construct as
many other points as may be desired.)

6. Given three points on the conic, and the tangent at two of them, to
construct the conic.

7. Given five points, two of which are at infinity in different
directions, to construct the conic. (In this, and in the following
examples, the student is supposed to be able to draw a line parallel to a
given line.)

8. Given four points on a conic (two of which are at infinity and two in
the finite part of the plane), together with the tangent at one of the
finite points, to construct the conic.

9. The tangents to a curve at its infinitely distant points are called
its _asymptotes_ if they pass through a finite part of the plane. Given
the asymptotes and a finite point of a conic, to construct the conic.

10. Given an asymptote and three finite points on the conic, to determine
the conic.

11. Given four points, one of which is at infinity, and given also that
the line at infinity is a tangent line, to construct the conic.

CHAPTER V - PENCILS OF RAYS OF THE SECOND ORDER

*79. Pencil of rays of the second order defined.* If the corresponding
points of two projective point-rows be joined by straight lines, a system
of lines is obtained which is called a pencil of rays of the second order.
This name arises from the fact, easily shown (§ 57), that at most two
lines of the system may pass through any arbitrary point in the plane. For
if through any point there should pass three lines of the system, then
this point might be taken as the center of two projective pencils, one
projecting one point-row and the other projecting the other. Since, now,
these pencils have three rays of one coincident with the corresponding
rays of the other, the two are identical and the two point-rows are in
perspective position, which was not supposed.

[Figure 19]

FIG. 19

*80. Tangents to a circle.* To get a clear notion of this system of
lines, we may first show that the tangents to a circle form a system of
this kind. For take any two tangents, _u_ and _u'_, to a circle, and let
_A_ and _B_ be the points of contact (Fig. 19). Let now _t_ be any third
tangent with point of contact at _C_ and meeting _u_ and _u'_ in _P_ and
_P'_ respectively. Join _A_, _B_, _P_, _P'_, and _C_ to _O_, the center of
the circle. Tangents from any point to a circle are equal, and therefore
the triangles _POA_ and _POC_ are equal, as also are the triangles _P'OB_
and _P'OC_. Therefore the angle _POP'_ is constant, being equal to half
the constant angle _AOC + COB_. This being true, if we take any four
harmonic points, _P__1_, _P__2_, _P__3_, _P__4_, on the line _u_, they
will project to _O_ in four harmonic lines, and the tangents to the circle
from these four points will meet _u'_ in four harmonic points, _P'__1_,
_P'__2_, _P'__3_, _P'__4_, because the lines from these points to _O_
inclose the same angles as the lines from the points _P__1_, _P__2_,
_P__3_, _P__4_ on _u_. The point-row on _u_ is therefore projective to the
point-row on _u'_. Thus the tangents to a circle are seen to join
corresponding points on two projective point-rows, and so, according to
the definition, form a pencil of rays of the second order.

*81. Tangents to a conic.* If now this figure be projected to a point
outside the plane of the circle, and any section of the resulting cone be
made by a plane, we can easily see that the system of rays tangent to any
conic section is a pencil of rays of the second order. The converse is
also true, as we shall see later, and a pencil of rays of the second order
is also a set of lines tangent to a conic section.

*82.* The point-rows _u_ and _u'_ are, themselves, lines of the system,
for to the common point of the two point-rows, considered as a point of
_u_, must correspond some point of _u'_, and the line joining these two
corresponding points is clearly _u'_ itself. Similarly for the line _u_.

*83. Determination of the pencil.* We now show that _it is possible to
assign arbitrarily three lines, __a__, __b__, and __c__, of __ the system
(besides the lines __u__ and __u'__); but if these three lines are chosen,
the system is completely determined._

This statement is equivalent to the following:

_Given three pairs of corresponding points in two projective point-rows,
it is possible to find a point in one which corresponds to any point of
the other._

We proceed, then, to the solution of the fundamental

PROBLEM. _Given three pairs of points, __AA'__, __BB'__, and __CC'__, of
two projective point-rows __u__ and __u'__, to find the point __D'__ of
__u'__ which corresponds to any given point __D__ of __u__._

[Figure 20]

FIG. 20

On the line _a_, joining _A_ and _A'_, take two points, _S_ and _S'_, as
centers of pencils perspective to _u_ and _u'_ respectively (Fig. 20). The
figure will be much simplified if we take _S_ on _BB'_ and _S'_ on _CC'_.
_SA_ and _S'A'_ are corresponding rays of _S_ and _S'_, and the two
pencils are therefore in perspective position. It is not difficult to see
that the axis of perspectivity _m_ is the line joining _B'_ and _C_. Given
any point _D_ on _u_, to find the corresponding point _D'_ on _u'_ we
proceed as follows: Join _D_ to _S_ and note where the joining line meets
_m_. Join this point to _S'_. This last line meets _u'_ in the desired
point _D'_.

We have now in this figure six lines of the system, _a_, _b_, _c_, _d_,
_u_, and _u'_. Fix now the position of _u_, _u'_, _b_, _c_, and _d_, and
take four lines of the system, _a__1_, _a__2_, _a__3_, _a__4_, which meet
_b_ in four harmonic points. These points project to _D_, giving four
harmonic points on _m_. These again project to _D'_, giving four harmonic
points on _c_. It is thus clear that the rays _a__1_, _a__2_, _a__3_,
_a__4_ cut out two projective point-rows on any two lines of the system.
Thus _u_ and _u'_ are not special rays, and any two rays of the system
will serve as the point-rows to generate the system of lines.

*84. Brianchon's theorem.* From the figure also appears a fundamental
theorem due to Brianchon:

_If __1__, __2__, __3__, __4__, __5__, __6__ are any six rays of a pencil
of the second order, then the lines __l = (12, 45)__, __m = (23, 56)__,
__n = (34, 61)__ all pass through a point._

[Figure 21]

FIG. 21

*85.* To make the notation fit the figure (Fig. 21), make _a=1_, _b = 2_,
_u' = 3_, _d = 4_, _u = 5_, _c = 6_; or, interchanging two of the lines,
_a = 1_, _c = 2_, _u = 3_, _d = 4_, _u' = 5_, _b = 6_. Thus, by different
namings of the lines, it appears that not more than 60 different
_Brianchon points_ are possible. If we call 12 and 45 opposite vertices of
a circumscribed hexagon, then Brianchon's theorem may be stated as
follows:

_The three lines joining the three pairs of opposite vertices of a hexagon
circumscribed about a conic meet in a point._

*86. Construction of the pencil by Brianchon's theorem.* Brianchon's
theorem furnishes a ready method of determining a sixth line of the pencil
of rays of the second order when five are given. Thus, select a point in
line 1 and suppose that line 6 is to pass through it. Then _l = (12, 45)_,
_n = (34, 61)_, and the line _m = (23, 56)_ must pass through _(l, n)_.
Then _(23, ln)_ meets 5 in a point of the required sixth line.

[Figure 22]

FIG. 22

*87. Point of contact of a tangent to a conic.* If the line 2 approach as
a limiting position the line 1, then the intersection _(1, 2)_ approaches
as a limiting position the point of contact of 1 with the conic. This
suggests an easy way to construct the point of contact of any tangent with
the conic. Thus (Fig. 22), given the lines 1, 2, 3, 4, 5 to construct the
point of contact of _1=6_. Draw _l = (12,45)_, _m =(23,56)_; then _(34,
lm)_ meets 1 in the required point of contact _T_.

[Figure 23]

FIG. 23

*88. Circumscribed quadrilateral.* If two pairs of lines in Brianchon's
hexagon coalesce, we have a theorem concerning a quadrilateral
circumscribed about a conic. It is easily found to be (Fig. 23)

_The four lines joining the two opposite pairs of vertices and the two
opposite points of contact of a quadrilateral circumscribed about a conic
all meet in a point._ The consequences of this theorem will be deduced
later.

[Figure 24]

FIG. 24

*89. Circumscribed triangle.* The hexagon may further degenerate into a
triangle, giving the theorem (Fig. 24) _The lines joining the vertices to
the points of contact of the opposite sides of a triangle circumscribed
about a conic all meet in a point._

*90.* Brianchon's theorem may also be used to solve the following
problems:

_Given four tangents and the point of contact on any one of them, to
construct other tangents to a conic. Given three tangents and the points
of contact of any two of them, to construct other tangents to a conic._

*91. Harmonic tangents.* We have seen that a variable tangent cuts out on
any two fixed tangents projective point-rows. It follows that if four
tangents cut a fifth in four harmonic points, they must cut every tangent
in four harmonic points. It is possible, therefore, to make the following
definition:

_Four tangents to a conic are said to be harmonic when they meet every
other tangent in four harmonic points._

*92. Projectivity and perspectivity.* This definition suggests the
possibility of defining a projective correspondence between the elements
of a pencil of rays of the second order and the elements of any form
heretofore discussed. In particular, the points on a tangent are said to
be _perspectively related_ to the tangents of a conic when each point lies
on the tangent which corresponds to it. These notions are of importance in
the higher developments of the subject.

[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._

[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:

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