How to find a harmonic conjugate based on a complete quadrangle determined by a triangle and a point inside it?

Question

How could you find a harmonic conjugate of points ( where: $(A,B,C,D) = \frac {AC}{AD}.\frac {BD}{BC} = -1 $ )

based on a complete quadrangle determined by a triangle $\triangle$ PQR and a point inside it?

Answer 1

Consider the first figure below where three points $A,B,C$ are given on the same straight line. The task is to determine the point $D$ so that that the cross ratio $$(AB;CD)=-1. \tag 1$$

Let $P=A$, $Q=B$ and $R$ be an arbitrary point not on the staright line determined by $AB$. Also, let $S$ be an arbitrary point inside $PQR$ as shown in the second figure.

Now, join $C$ and $S$ with a straight line and denote the intersection points of this straight line and the segment $PR$ and the segment $QR$ by $E$ and $F$, respectively. Connect $AF$ and $BE$ with straight lines and denote the intersection point by $G$.

Finally, the intersection point of $GR$ and $AB$ will be a point $D$ so that $(1)$ holds.