Conic pencils and inversion (Q1591212)

Let \(D_0,D_1,D_2\) be the diagonal points of a complete quadrangle \(Q\) (vertices \(V_1,\dots, V_4)\) in a Pappian projective plane \(P^2\) of characteristic \(\neq 2\). The conjugacy mapping relative to \(Q\) is oonstructed by using a bijection \(u\mapsto C_u\) of the set of lines \(\{u\}\) onto the bundle of conics \(\{C\}\) containing \(D_0,D_1,D_2\): the conic \(C_u\) consists of all points of \(P^2\) having \(u\) as the corresponding polar line with respect to some conic going through \(V_1,\dots,V_4\). Applying this to Euclidean geometry, the inversion with center \(O\) of the inversion circle going through the point \(P\) in the Euclidean plane is proved to be the product of the reflection at \(OP\) and the affine restriction of the conjugacy mapping relative to the quadrangle having \(P\) as one of its vertices and \(O\) together with the circular points at infinity as diagonal points.