Here we study the system of perspectivities naturally linked to a four-point.

[1] The (g,G)-perspectivity is the involutive homography S fixing the points of line g and the point G. Besides, each point I is mapped to J lying on line GI and such that {I,J} are harmonic conjugate to points G, G

[2] The (f,F)-perspectivity is the involutive homography T fixing the points of line f and the point F. Besides, each point I is mapped to K lying on line FI and such that {I,K} are harmonic conjugate to points F, F

[3] The two maps commute: S*T = T*S = R and define a third perspectivity, the (e,E)-perspectivity, fixing the points of line e etc..

[4] R

The commutativity S*T = T*S generates for every point I of the plane another four-point and all these have the same diagonal points {E,F,G} and the same diagonals {e,f,g}. Considering the group generated by S and T, triangle EFG is its

Considering the figure as part of the projective plane, triangle EFG defines four other triangles, each having a common side with the "central" one and a common vertex, the four triangles covering completely the plane. The maps {S, T, R} interchange these triangles with the "central" one.

Each of these triangles is also

See the file FourPointsCyclic.html for an illustration of the case of an obtuse triangle EFG and the corresponding unique, invariant under the group, circle.

In general the product of two involutions is a non involutive homography. The commutativity is equivalent with the condition that the Fregier point of each is contained in the homography axis of the other. This exactly is the case with the three involutions connected with a four-point.

FourPointsCyclic.html

FregierInvolutionsProduct.html

Harmonic_Perspectivity.html

HomographyAxis.html

InvolutionsProduct.html

InvolutionsProductGeneral.html

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