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COURSE OF PROJECTIVE GOMETRY

§ 3: answers

*O8* Desargues: given two triangles *a*_{1}*b*_{1}*c*_{1} and
*a*_{2}*b*_{2}*c*_{2}, where *a*_{1}.*a*_{2}, *b*_{1}.*b*_{2} and
*c*_{1}.*c*_{2} are lying on one line (then we call the triangles *line perspective*); then the connecting lines
*a*_{1}*b*_{1}-*a*_{2}*b*_{2},
*a*_{1}*c*_{1}-*a*_{2}*c*_{2} and
*b*_{1}*c*_{1}-*b*_{2}*c*_{2} concur (we call the triangles correspondingly *point perspective*).

Notice that the dual theorem of Desargues is the reverse of the theorem of Desargues: together they form the theorem that two triangles are point perspective if and only if they are (correspondingly)
line perspective.

Pappus: Given two points *L* and *M*, three lines *a*_{1},*a*_{2},*a*_{3} through *L*, and
three lines *b*_{1},*b*_{2},*b*_{3} through *M*. Then the connecting lines
*a*_{1}*b*_{2}-*a*_{2}*b*_{1},
*a*_{1}*b*_{3}-*a*_{3}*b*_{1} and
*a*_{3}*b*_{2}-*a*_{2}*b*_{3} concur.

*O9* Using Desargues: choose two points *A*_{1} and *B*_{1} on *l* and two points *A*_{2} and *B*_{2} on *m*, so that
*l*.*m* = *A*_{1}*B*_{1}. *A*_{2}*B*_{2} = *T*.

Make a Desargues configuration with triangles *A*_{1}*B*_{1}*C*_{1} and *A*_{2}*B*_{2}*C*_{2} that are point perspective
with center *T*, so that *P* = *A*_{1}*C*_{1}. *A*_{2}*C*_{2}.

Then the required line goes through *P* and *B*_{1}*C*_{1}. *B*_{2}*C*_{2}.

*O10* Using Pappus: draw a line *m* that doesn't coincide with *l* and not through *A* or *B*. Choose points *L*_{1},*L*_{2},*L*_{3}
on *l* and *M*_{1},*M*_{2},*M*_{3} on *m* so that *A* = *L*_{1}*M*_{2}. *L*_{2}*M*_{1},
*B* = *L*_{2}*M*_{3}. *L*_{3}*M*_{2}. Then the line through *A* and *B* is the line through *B* and
*L*_{1}*M*_{3}. *L*_{3}*M*_{1}.

General method: first make a configuration of Desargues or (dual) Pappus, and find therein three points on a line or three lines through a point, whatever the problem requires.

Now make the construction by adding an element of the configuration again and again, as the problem requires, checking each time whether some elements of the configuration are now fixed or
that they all can be chosen freely.

*O11*

*A**B**C* and *A*'*B*'*C*' are point perspective, but not line perspective.

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