There is a basic figure by which we define involutive homographies on lines. This is done by intersecting a line (e) with the six sides of a complete quadrilateral. The three pairs of points (X1,X2), (Y1,Y2) and (Z1,Z2) intercepted on (e) by corresponding pairs of opposite sides of the quadrangle are in involution. This is discussed in InvolutionBasic.html . Here we consider the three circles with diameters the segments (X1X2), (Y1Y2) and (Z1Z2). They belong to the same circle bundle. Consequently there are two points H1, H2 (real or imaginary), with the property: each of the three pairs (X1X2), (Y1Y2) and (Z1Z2) is harmonic conjugate with respect to H1 and H2. Here is the picture of the case in which the circle bundle is of intersecting type, hence points H1, H2 are complex points.
The case of circle bundle of non-intersecting type, and consequently real points H1, H2 is discussed in InvolutionBasic2.html .
By the way, these are exactly the configurations of four points {A,B,C,D} and line (e) for which there is no conic passing through the four points and tangent to (e). Look at FourPtsAndTangent.html for a discussion on that.
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Line Involution on a line and circle bundle generated - II ![[0_0]](InvolutionBasic3_files/InvolutionBasic3_0_0_0.png)
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