Polar3

Preservation of cross ratio by polarity

Consider a circle (c), a point E and its polar line (e) with respect to the circle. The cross ratio of four points A, B, C, D on line (e) is the same with the cross ratio of the line bundle of the corresponding polars a, b, c, d.

The result is a consequence of the properties of the cross ratio (see the file Harmonic_Bundle.html ). The cross ratio of the bundle of lines EA, EB, EC, ED is the same with the cross ratio of points A, B, C, D. Besides this cross ratio is determined through the sines of the angles of the lines. But lines EA, EB, etc. are orthogonal to the corresponding lines a, b, .... Thus angle(AEB) = angle(a,b), etc. and the cross ratio of the two line bundles are identical.

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