An Elementary Course in Synthetic Projective Geometry Part 9

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 circ.u.mscribed 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 circ.u.mscribed 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, pa.s.ses 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.'_