COURSE OF PROJECTIVE GEOMETRY

§ 8: answers

*O32*

*O33*

*l*(*A*,*B*,*C*) _{∧}=^{Q} *m*(*A*,*C'*,*B'*) _{∧}=^{P} *l*(*A*,*C*,*B*)

*L*(*a*,*b*,*c*) _{∧}=^{q} *M*(*a*,*c'*,*b'*) _{∧}=^{p} *L*(*a*,*c*,*b*)

*O34*

i) *A*=*A'*. Then we can take the perspectivity from *l* onto *m* with center *BB'.CC'*.

ii) The six points are distinct and *l* ≠ *m*. See the drawing.

Choose *n* through *A'* and *P* on *AA'*.
*B"* and *C"* are the projections of *B* and *C* from *P* on *n*.

Then we can take the composition φoψ where ψ is the perspectivity from *l* onto *n* with center *P*, and φ the perspectivity from *n* onto *m* with center
*B'B".C'C"*.

iii) The six points are lying on one line, so *l*=*m*.

Study the picture above and show we can take a projectivity that is the product of three perspectivities.