To prove F(Z

(Y

But the claimed equality can be read off from the figure.

(Y

(Y

The property can be extended to every conic passing through {A,B,C,D}. In fact, the set of two lines through G is a degenerate conic (passing through the four points) and its intersection points {Z

There is another aspect of the involution on (e) defined by the quadrilateral, connected with a circle bundle defined by the three circles with diameters X

In fact, construct an arbitrary triangle ABD whose sides pass through {X

The three sides {a,b,c} of ABC and the three lines {a',b',c'} emanating from the vertices of ABC (as shown below) coincide in a point D precisely when lines {a,a',b,b',c,c'} intersect an arbitrary line (e) not passing through points {A,B,C,D} at points {X

The

The inverse (

Assume the condition is satisfied and lines {a',b'} (only the two) intersect at D. Draw then through D line c''=AD and apply the necessity proof to {a',b',c''} and their intersection points {X

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