whence

_The abscissas of two points on a parabola are to each other as the squares of the corresponding coordinates, a diameter and the tangent to the curve at the extremity of the diameter being the axes of reference._

The last equation may be written

_y__2__ = 2px,_

where _2p_ stands for _y'__2__ : x'_.

The parabola is thus identified with the curve of the same name studied in treatises on a.n.a.lytic geometry.

*120. Equation of central conics referred to conjugate diameters.*

Consider now a _central conic_, that is, one which is not a parabola and the center of which is therefore at a finite distance. Draw any four tangents to it, two of which are parallel (Fig. 31). Let the parallel tangents meet one of the other tangents in _A_ and _B_ and the other in _C_ and _D_, and let _P_ and _Q_ be the points of contact of the parallel tangents _R_ and _S_ of the others. Then _AC_, _BD_, _PQ_, and _RS_ all meet in a point _W_ (-- 88). From the figure,

_PW : WQ = AP : QC = PD : BQ,_

or

_AP BQ = PD QC._

If now _DC_ is a fixed tangent and _AB_ a variable one, we have from this equation

_AP BQ = __constant._

This constant will be positive or negative according as _PA_ and _BQ_ are measured in the same or in opposite directions. Accordingly we write

_AP BQ = b__2__._

[Figure 31]

FIG. 31

Since _AD_ and _BC_ are parallel tangents, _PQ_ is a diameter and the conjugate diameter is parallel to _AD_. The middle point of _PQ_ is the center of the conic. We take now for the axis of abscissas the diameter _PQ_, and the conjugate diameter for the axis of ordinates. Join _A_ to _Q_ and _B_ to _P_ and draw a line through _S_ parallel to the axis of ordinates. These three lines all meet in a point _N_, because _AP_, _BQ_, and _AB_ form a triangle circ.u.mscribed to the conic. Let _NS_ meet _PQ_ in _M_. Then, from the properties of the circ.u.mscribed triangle (-- 89), _M_, _N_, _S_, and the point at infinity on _NS_ are four harmonic points, and therefore _N_ is the middle point of _MS_. If the coordinates of _S_ are _(x, y)_, so that _OM_ is _x_ and _MS_ is _y_, then _MN = y/2_. Now from the similar triangles _PMN_ and _PQB_ we have

_BQ : PQ = NM : PM,_

and from the similar triangles _PQA_ and _MQN_,

_AP : PQ = MN : MQ,_

whence, multiplying, we have

_b__2__/4 a__2__ = y__2__/4 (a + x)(a - x),_

where

[formula]

or, simplifying,

[formula]

which is the equation of an ellipse when _b__2_ has a positive sign, and of a hyperbola when _b__2_ has a negative sign. We have thus identified point-rows of the second order with the curves given by equations of the second degree.

PROBLEMS

1. Draw a chord of a given conic which shall be bisected by a given point _P_.

2. Show that all chords of a given conic that are bisected by a given chord are tangent to a parabola.

3. Construct a parabola, given two tangents with their points of contact.

4. Construct a parabola, given three points and the direction of the diameters.

5. A line _u'_ is drawn through the pole _U_ of a line _u_ and at right angles to _u_. The line _u_ revolves about a point _P_. Show that the line _u'_ is tangent to a parabola. (The lines _u_ and _u'_ are called normal conjugates.)

6. Given a circle and its center _O_, to draw a line through a given point _P_ parallel to a given line _q_. Prove the following construction: Let _p_ be the polar of _P_, _Q_ the pole of _q_, and _A_ the intersection of _p_ with _OQ_. The polar of _A_ is the desired line.

CHAPTER VIII - INVOLUTION

[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,