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,

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._