PROJECTIVE GEOMETRY COURSE

*Extra problems*

1. *K* is a non-degenerated conic through two points *A* and *B*, and the line *l* intersects *K* in two other points *C* and *D*.
For *X* on *l*, let *S _{X}* be the "second" intersection point of

a) Prove that the mapping ψ :

b) Suppose

c) Make a sketch from which it becomes evident that ψ isn't identity in all cases.

d) Prove that ψ is hyperbolic if it's not identity.

e) Suppose we have on

f) Show how we can construct

g) Take the situation given in b), except

2. Given a non-degenerated conic *K* and two lines *l* and *m*, whilst the pole of *m* doesn't lie on *l*. We map each point *X* on *l* to
π(*X*) = *x* **.** *m*, where *x* is the polar line of *X* with respect to *K*.

Prove that π is a projectivity from *l* onto *m*, and that the Pappus line of π goes through the poles *L* of *l* and *M* of *m*.

3. Given five freely situated points and a line that doesn't go through any of these five points. Construct all intersection points (if there are any) of the line and the conic through the five
points.

(Hint: consider the projectivity *x* **.** *l* → φ(*x*) **.** *l*, where φ is a projectivity between pencils of lines that generates the conic.)

4. Given four proper points on a parabola and the direction of its axis. Construct its top.

5. Given two parallel lines *l* and *m* and four points *A*, *B*, *C*, *D*, whilst *A* and *B* don't lie on *l* and *C* and *D* don't lie on
*m*.

Construct a point *X* on *l* and a point *Y* on *m* such that *AX* is parallel to *CY* and *BX* is parallel to *DY*.

(Hint: first choose a suitable projectivity from *l* onto itself.)

6. *A* and *B* are two points of a conic *K*. The tangents to *K* in *A* and *B* intersect each other in *S*. An arbitrary line through *S* intersects
*K* in *C* and *D*. The tangents *SA* and *SB* intersect the tangent in *C* in the intersection points *E* and *F*; the lines *DA* and *DB* intersect
this last tangent in *G* and *H*. Finally, *L* is the intersection point of *AB* and this tangent in *C*.

Show that *L* and *C* separate harmonically with both the pairs {*E*, *F*} and {*G*, *H*}. Prove that the tangent to *K* in *D* is also tangent to the
conic through *A*, *B*, *D*, *E* and *F*.

7. In a plane the points *A*, *B*, *P* and *Q* are given. *A* and *B* don't lie on *PQ*. We consider the translation φ over vector __q__-__p__.

What is the geometric locus of the intersection points *AX* **.** *B*φ(*X*) with *X* on *PQ*?

8. Given the parabola x_{2}x_{3} = x_{1}^{2}. Determine the projective coordinates of the polar line of the top, the polar line of the point at infinity on the axis,
and the pole of the axis.

9. Of a conic *K* five points *A*, *B*, *C*, *S*, *T* are given. A conic *K* ' goes through *A*, *B*, *C*, *U*, *V*.
Construct, if possible, an intersection point of *K* and *K* ' that's not *A*, *B* or *C*. Which cases do occur?

(Hint: consider a suitable projectivity φ between pencils of lines that generates *K* and a suitable projectivity ψ between pencils of lines that generates *K* '.)

10. A variable conic *K* has four fixed tangents *a*, *b*, *c*, *d*. From a fixed point *A* on *a* we draw the other tangent α to
*K*, and from a fixed point *D* on *d* the other tangent δ to *K*. Prove that α **.** δ runs through a fixed straight line.

(Hint: Brianchon.)

11. Through a point *O* we draw four lines that intersect a conic *K* in (respectively) *A* and *A*_{1}, *B* and *B*_{1}, *C* and
*C*_{1}, *D* and *D*_{1}. *AB* and *CD* intersect each other in *L*, *A*_{1}*B*_{1} and *C*_{1}*D*_{1}
intersect each other in *M*.

Prove that *O*, *L* and *M* are collinear.

12. Given the triangle *ABC* and the line *l*. Suppose a variable conic has *l* and the sides of the triangle among its tangents. The contact points on *BC*, *AC* and *AB* are
*A*_{1}, *B*_{1} and *C*_{1}, respectievely. Prove:

a) *AA*_{1}, *BB*_{1} and *CC*_{1} go through one and the same point *P*.

b) The pencils (*A*_{1}) and (*B*_{1}) of lines are perspective.

c) The variable point *P* lies on a fixed conic through *A*, *B* and *C*.