PROJECTIVE GEOMETRY COURSE

§ 25: *The relation between conics of points and conics of lines*

*Proposition:* The tangents to a conic of points form a conic of lines (and the points of contact to a conic of lines form a conic of lines).

*Proof:* Let *l*, *m*, *n* be tangents to a conic of points *J*, with contact points *L*, *M*, *N*.
Let *X* := *l***.***m*, *Y* := *m***.***n*, *Z* := *l***.***n*.

The projectivity φ : *l* → *m* with φ(*X*) = *M*, φ(*L*) = *X* and φ(*Z*) = *Y* has Pappus line *p* = *LM*.

Now let *t* be a fourth tangent, with contact point *T*.

Let *U* := *l***.***t*, *V* := *m***.***t*, *Q* := *p***.***TN*.

Then *U* and *Y* lie on the diagonal through *Q* and *TM***.***NL* of quadrangle *TMNL* (zie §22).

Then *Y*, *U*, *Q* are collinear, and likewise *V*, *Q*, *Z*.

According to Steiner, φ maps *U* to *V*. So *t* is the line connecting *U* and φ(*U*).

So each tangent to *J* is a line connecting a point on *l* and its image point under φ on *m*.

*Problems:*

*O92* Suppose we have five tangents *a*, *b*, *c*, *d*, *e* to a conic of points *J*. Construct the pole of a given line *x*.

(Hint: if *a '* is the second tangent to *J* through *a***. ***x*, then the pole *X* of *x* lies on the fourth harmonic *x '* with {*a*, *a '* } and
*x*. Prove this first.)

*O93* Let *B* and *C* be conjugated points with respect to a conic of points *J*. Suppose we have a line through *C* which intersects *J* in *P* and *Q*.
Suppose *BP* and *BQ* intersect *J* in two more points *R* and *S*.

Prove that *C*, *R* and *S* are collinear.

*O94* Prove that a quadrangle *ABCD* with *A*, *B*, *C*, *D* on *J* and the quadrilateral *abcd* of the tangents to *J* in *A*, *B*, *C*,
*D*, respectively, have the same diagonal triangle.