An Elementary Course in Synthetic Projective Geometry Part 10

*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 circ.u.mscribed 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 pa.s.sing 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:

The polar line of an infinitely distant point is called a _diameter_, and the pole of the infinitely distant line is called the _center_, of the conic.

*108.* From the harmonic properties of poles and polars,

_The center bisects all chords through it (-- 39)._

_Every diameter pa.s.ses through the center._

_All chords through the same point at infinity (that is, each of a set of parallel chords) are bisected by the diameter which is the polar of that infinitely distant point._

*109. Conjugate diameters.* We have already defined conjugate lines as lines which pa.s.s each through the pole of the other (-- 100).

_Any diameter bisects all chords parallel to its conjugate._

_The tangents at the extremities of any diameter are parallel, and parallel to the conjugate diameter._

_Diameters parallel to the sides of a circ.u.mscribed parallelogram are conjugate._

All these theorems are easy exercises for the student.

*110. Cla.s.sification of conics.* Conics are cla.s.sified according to their relation to the infinitely distant line. If a conic has two points in common with the line at infinity, it is called a _hyperbola_; if it has no point in common with the infinitely distant line, it is called an _ellipse_; if it is tangent to the line at infinity, it is called a _parabola_.

*111.* _In a hyperbola the center is outside the curve_ (-- 101), since the two tangents to the curve at the points where it meets the line at infinity determine by their intersection the center. As previously noted, these two tangents are called the _asymptotes_ of the curve. The ellipse and the parabola have no asymptotes.

*112.* _The center of the parabola is at infinity, and therefore all its diameters are parallel,_ for the pole of a tangent line is the point of contact.

_The locus of the middle points of a series of parallel chords in a parabola is a diameter, and the direction of the line of centers is the same for all series of parallel chords._

_The center of an ellipse is within the curve._

[Figure 28]

FIG. 28

*113. Theorems concerning asymptotes.* We derived as a consequence of the theorem of Brianchon (-- 89) the proposition that if a triangle be circ.u.mscribed about a conic, the lines joining the vertices to the points of contact of the opposite sides all meet in a point. Take, now, for two of the tangents the asymptotes of a hyperbola, and let any third tangent cut them in _A_ and _B_ (Fig. 28). If, then, _O_ is the intersection of the asymptotes,-and therefore the center of the curve,- then the triangle _OAB_ is circ.u.mscribed about the curve. By the theorem just quoted, the line through _A_ parallel to _OB_, the line through _B_ parallel to _OA_, and the line _OP_ through the point of contact of the tangent _AB_ all meet in a point _C_. But _OACB_ is a parallelogram, and _PA = PB_.