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COURSE OF PROJECTIVE GEOMETRY

§ 23: *Conics of points and conics of lines*

*Definition:* A non-degenerated *conic of points* is the set of intersection points *x*** . **x' , where *x* → *x'* is a projectivity (but not a perspectivity)
from a pencil of lines *L* onto a pencil of lines *M* (with *L* ≠ *M*).

*Note:* Each projectivity φ : *L* → *M* is the restriction to *L* of a projective transformation of *P*^{2}. So this definition is equivalent
to the one in §21.

*Definition:* A non-degenerated *conic of lines * is the set of connecting lines *XX '* , where *X* → *X '* is a projectivity (but not a perspectivity)
from a pencil of points *l* onto a pencil of points *m* (with *l* ≠ *m*).

*Note:* It will turn out (in §25) that a conic of lines is the set of tangents to a conic of points.

*Example:*

Given *a*, *b*, *c* through *L* and *a '* , *b '* , *c '* through *M*. Construct the points of the conic that corresponds to the projectivity
from *L* to *M* which maps *a* to *a '* , *b* to *b '* , and *c* to *c '* .

*Solution:*

First construct the Pappus point *P* of the projectivity.

Let *x* be a line through *L*. Construct *x'* using the construction of Steiner. See *O45*.

The intersection point *x*** . **x' is a new point of the conic.

*Note:* (See *O43*.)

If *x* = *LM* then *x ' * = *MP* so *x*** . **x' = *M*.

If *x* = *LP* then *x ' * = *LM* so *x*** . **x' = *L*.

So *L* and *M* belong to the conic.

*Example:*

Given five freely situated points (that is to say: no three collinear). Construct a sixth point of the conic through these five points.

*Solution:*

Call two of these points *L* and *M*, and the other three *A*, *B* and *C*.

Let *a* := *LA*, *b* := *LB*, *c* := *LC*, and let *a '* := *MA*, *b '* := *MB*, *c '* := *MC*.

Let φ be the projectivity from *L* to *M* with φ(*a*) = *a '* , φ(*b*) = *b '*, φ(*c*) = *c '*.

Then φ generates the required conic.

Construct a sixth point as in the last example above.

*O84* Investigate which degenerations occur if in the definition of a conic of points *L* = *M*, or if the projectivity is a perspectivity. Likewise for a conic of lines.

*O85* Let *A*, *B* and *C* be points on a line *l* and let *A ' *, *B ' * and *C ' * be points on a line *m*.

Let φ be the projectivity from *l* onto *m* with φ(*A*) = *A '* , φ(*B*) = *B '*, φ(*C*) = *C '*.

Construct the lines of the conic of lines generated by φ.

Furthermore, prove that *l* and *m* belong to this conic of lines.

*O86* Given five freely situated lines. Construct a sixth line of the conic of lines determined by these five. (For a good picture: start from five tangents to a circle.)

*O87* Prove a line can't have three distinct points in common with a non-degenerated conic (of points).

answers

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