[alogo] Polar of a point with respect to a circle

Consider a circle (c) a point A and the diacenter-line (a) intersecting (c) at G and J. Consider also an arbitrary point D on the circle (c) and the line b = [AD], intersecting again (c) at E. Extend GD and JE to their intersection point F. As D moves on the circle (c), the corresponding point F describes a line (red). This line is the [polar] line of C with respect to the circle (c). The discussion below is a variation on the subject handled in CyclicProjective.html .

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1) Per construction, triangle FGJ has altitudes DJ, GE intersecting at a point H. Hence its third altitude FI passes also trhough H and contains the middle K of FH.
2) The identification of line FH with the polar of C follows from the fact, that points G, J, in our figure, are indeed harmonic conjugate to A and I (see the basic picture in Harmonic.html ). Hence the location of I is uniquely defined on line a. Besides line FH defines on every secant line b of the circle through A a point T, which harmonic conjugate to A with respect to A (see reference cited above). But this is the characteristic property of the polar of A.
3) Triangles FDJ, BEK are similar orthogonal triangles, since angle(KBE) = angle(DJE) and angle(BKE) = angle(DFE). Hence circles (d) and (c) are orthogonal.
4) It follows that circle (d) is orthogonal to every circle-member of the circle-bundle (I), consisting of circles passing through the intersection points L, M of (c) with the polar line KH.
5) All circles (d) belong to the circle bundle (II) which is the orthogonal bundle of (I).
6) Consider an arbitrary circle-member (e) of bundle (I), tracing on line (a) the diameter OP. The corresponding triangle OFP has again H as its orthocenter. Indeed, consider an intersection point Q of circles (e) and (d). angle(FQH) is a right one, hence line QH passes through P, and PQ is an altitude of OFP, passing through H.


Notice that the orthocenter H* of triangle QSF (similar to PFO) is the symmetric of H with respect to the middle R of the common chord QS of circles d and e. The locus of H* is a circle tangent to d at F. For this story see the CircleBundleLocus.html .

A figure somewhat overloaded with drawings on the bundles (I) and (II) containing all circles (d), connected to these remarks, can be found in Polar5.html .

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