PolarLocus

Polar of a point with respect to a circle (2)

The polar line of a point A with respect to a circle (c) can be described as the locus of orthocenters of certain triangles constructed by secants of the circle (c). In fact, it is the line (red) containing the orthocenters H of triangles GCD, constructed by intersecting CE and DF, for a variable secant AF of circle (c) through the point A (see Polar.html ). The other triangle EFG is similar to GCD and its orthocenter I is symmetric to H with respect to the middle of side EF. The locus of I, for the various secants [AEF] through A is the curve shown.

The curve is restricted in a strip, symmetric with respect to the center B of circle c. It has three self intersections, is symmetric with respect to the line a and has an asymptotic line.
Another variant of this subject and a simpler locus for the orthocenter H, when G remains constant and varies circle c, is discussed in the file CircleBundleLocus.html .

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