PROJECTIVE GEOMETRY COURSE

§ 24: *Tangents and points of contact*

*Proposition:* Let *J* be a non-degenerated conic of points. Then for each point *A* of *J* there is exactly one line through *A* that isn't incident with any other point
of *J*.

*Proof:* Take besides *A* four other points of *J*, say *B*, *C*, *D* and *E*.

Then *J* is the conic of points determined by the projectivity φ from *A* onto *B* that maps *AC* to *BC*, *AD* to *BD*, and
*AE* to *BE*.

Let *P* be the Pappus point of this projectivity. Each line *x* through *A* has with *J* the second point *x* **.** φ(*x*) in common.

If *x* = *AP* then φ(*x*) = *AB* and *x* **.** φ(*x*) = *A*. So *AP* is a line as required.

If *x* ≠ *AP* then φ(*x*) ≠ *AP* (for φ is 1-1), so then φ(*x*) doesn't go through *A* (for φ(*x*) does go through *B*), and then
*x* **.** φ(*x*) ≠ *A*.

So *AP* is the only line that meets the requirements.

*Definition:* We call the line through *A* on *J* that isn't incident with any other point of *J* *tangent* to *J* in *A*.

*Note:* The proof of the last proposition gives a procedure for the construction of the tangent to *J* in *A* whenever five points of *J* are given among which *A*.

*Note:* In the situation of the proof of the last proposition, *BP* is the tangent to *J* in *B*. Prove this.

*Proposition:* Let *j* be a non-degenerated conic of lines. Then for each line *a* of *j* there is exactly one point on *a* that isn't incident with any other line of
*j*.

*Proof:* Dualise the proof of the previous proposition.

*Definition:* We call the point on the line *a* of *j* that isn't incident with any other line of *j* the *point of contact* to *j* on *a*.

*Problems:*

*O88* Given five freely situated points *A*, *B*, *C*, *D*, *E*. Construct the tangent in *A* to the conic through these five points.

*O89* Given five freely situated lines *a*, *b*, *c*, *d*, *e*. Construct the point of contact on *a* to the conic of lines determined by these five lines.

*O90* Given three lines *a*, *b*, *c* of a conic of lines *j*, and the contact points *A* on *a* and *B* on *b*. Construct more lines on
*j* and the contact point on *c*.

*O91* Given three points *A*, *B*, *C* of a conic of points *J*, and the tangents *a* in *A* and *b* in *B*. Construct more points of
*J* and the tangent in *C*.

*Note:* It's even easier to solve the problems of this section after studying §26. See *O101*.