PROJECTIVE GEOMETRY COURSE

§ 11: *Line and point of Pappus*

*Theorem of Pappus :* Let *A*, *B*, *C* be three points on a line *l*, and
*A'*, *B'*, *C'* three points on an other line *m*. Then the points *AB'.BA'*, *AC'.CA'*, *BC'.CB'* lie on one line (the *Pappus line* of the sextet).

*Proof:* Look at the figure beneath:

We find (*B'*, *T*, *C*, *X*) _{∧}=^{B} (*B'*, *C'*, *P*, *A'*) _{∧}=^{A}
(*Y*, *S*, *C*, *A'*).

So the projectivity that maps *B'* to *Y*, *T* to *S* and *X* to *A'*, leaves *C* invariant. Hence, according to *O40*, it is a perspectivity.

So *B'Y*, *TS* and *XA'* concur. This is what we had to prove.

*Theorem of Steiner :* Let φ be a projectivity from a line *l* onto an other line *m*. Then all points *Xφ(Y).Yφ(x)* with *X*, *Y* on *l*,
are collinear (on the *Pappus line *of the projectivity).

*Proof:* Choose three points *A*, *B*, *C* on *l*, and let *A'*:=*φ(A)*, *B'*:=*φ(B)*, *C'*:=*φ(C)*. Let *p* be the Pappus line
of the sextet.

If γ is the perspectivity from *l* onto *p* with center *A'*, and δ from *p* onto *m* with center *A*, then φ = δoγ (because φ and
δoγ both map *A*, *B*, *C* into (respectively) *A'*, *B'*, *C'* , FS).

So for *X* on *l* we find *φ(X)* by first projecting *X* from *A'* onto *p*, and, subsequently, the image point from *A* onto *m* (*Steiner construction*).

Hence, if *X* and *Y* are arbitrary points on *l*, then *p* is the line through *XA'.Aφ(x)* and *YA'.Aφ(Y)*. Applying the theorem of Pappus on
*A*, *X*, *Y*, *A'*, *φ(X)*, *φ(Y)*, we find that *Xφ(Y).Yφ(x)* lies on *p*.

*O43* Let φ be a projectivity from a line *l* on an other line *m*. With the help of the theorem of Steiner, find the image point of *p.l*, and the image point of
*l.m*.

Show with the help of *O40* that φ is a perspectivity if and only if *l*, *m* and *p* go through one point. Give also the dual assertion.

*O44* Let φ be a projectivity from a line *l* onto an other line *m*. Suppose we know the Pappus line and a pair (*A, φ(A)*) of φ.

Check in which cases there is exactly one φ with these data. Give also the dual assertion.

*O45* Let *a*, *b*, *c* be three lines through a point *L* and *a'*, *b'*, *c'* three lines through an other point *M*.

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

Construct thet Pappus point of the projectivity.

Let *l* be a fourth line through *L*. Construct *φ(l)*.

*O46* Look at the following figure and study the text under it. Subsequently, prove the theorem of Desargues.

The triangles *A _{1}*

So we have to show that

So we seek a projectivity with (

Then this is a perspectivity because of

*O47*
In the situation of *O34*, construct for arbitrary *X* on *l* the image point *X '* with the help of Steiner.