PROJECTIVE GEOMETRY COURSE

§ 12: *Harmonic separation and the complete quadrangle*

*Definition :* Let *A*, *B*, *C*, *D* be four points on a line *l*. The pairs {*A*, *B*} and {*C*, *D*} separate *harmonically*
if (*A*, *B*; *C*, *D*} = -1.

Let *a*, *b*, *c*, *d* be four lines through a point *L*. The pairs {*a*, *b*} and {*c*, *d*} separate *harmonically*
if (*a*, *b*; *c*, *d*} = -1.

See *O48*.

*Definition :* A *complete quadrangle* is the figure formed by four freely situated points (that is: no three collinear) and their six connecting lines (the *sides*).
*Opposite* sides are sides whose intersection points is not a vertex. We call the intersection point of two opposite sides a *diagonal point*. The connecting line of two diagonal
points is called *diagonal*.

There are three diagonal points (prove this) and three diagonals.

*Proposition :* If two sides and two diagonals of a complete quadrangle go through one diagonal point, they form pairs that separate harmoically.

*Proof* : (see the picture)
*l*(*H*_{1}, *H*_{2}, *X '*, *D*_{3}) _{∧}=^{D2}
*m*(*H*_{3}, *H*_{4}, *X*, *D*_{3}) _{∧}=^{D1}
*l*(*H*_{2}, *H*_{1}, *X '*, *D*_{3}), so
(*H*_{1}, *H*_{2}; *X '*, *D*_{3}) = (*H*_{2}, *H*_{1}; *X '*, *D*_{3}).

If we call this number d, then d=1/d. So d=__+__1. Since the points are distinct, we get d=-1.

*Remark :* Strictly speaking, we don't need cross ratio to build up the projective geometry of the real projective plane. The complete quadrangle alone is already sufficient, but then we have to
formulate some axioms of order and continuity.

(See *Coxeter* : "The real projective plane".)

*Problems:*

*O48* Suppose (*X*_{1}, *X*_{2}; *X*_{3}, *X*_{4}) = -1. Find all permutations σ of {1,2,3,4} such that
(*X*_{σ(1)}, *X*_{σ(2)}; *X*_{σ(3)}, *X*_{σ(4)}) = -1.

*O49* Let *A*, *B*, *C* be three points on a line *l*. Use the figure of the complete quadrangle to *construct the fourth harmonic* with
{*A*, *B*} and *C* (this is *X* such that (*A*, *B*; *C*, *X*) = -1).

*O50* Let *A*_{0}*A*_{1}*A*_{2}*A*_{3} and *B*_{0}*B*_{1}*B*_{2}*B*_{3} be complete
quadrangles with
*A*_{2} = *B*_{2} , *A*_{3} = *B*_{3} and *D*_{1} :=
*A*_{0}*A*_{1} . *A*_{2}*A*_{3} =
*B*_{0}*B*_{1} . *B*_{2}*B*_{3} := *E*_{1}.

Define the other diagonal points as follows:
*D*_{2} := *A*_{0}*A*_{2} . *A*_{1}*A*_{3} ,
*E*_{2} := *B*_{0}*B*_{2} . *B*_{1}*B*_{3} ,
*D*_{3} := *A*_{0}*A*_{3} . *A*_{1}*A*_{2} ,
*E*_{3} := *B*_{0}*B*_{1} . *B*_{1}*B*_{2} .

Prove that *D*_{2}*D*_{3} , *E*_{2}*E*_{3} and *A*_{2}*A*_{3} concur.

*O51* Let *P* be a point that's not lying on a side of triangle *A**B**C*. Construct the line that intersects *BC*, *CA* and *AB* in *D*, *E*, *F*
respectively, in such a way that {*P*, *F*} and {*D*, *E*} separate harmonically. (See *O49*.)