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.

[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._