[alogo] Orthocycle

Start with a circle c(A,r=|AB|) and a point F inside this circle. Define the polar line (f) of F with respect to the circle. Define also the circle e(H,|HA|) with diameter the segment FA. This circle (e) is the inverse of the polar (f) of F with respect to (c). Consider now an arbitrary circle d(C, |CE|) of the coaxal bundle II, of circles orthogonal to all circles of bundle I(c,e), generated by (c) and (e). All these circles (d) of bundle II pass through a fixed point G (limit point of a coaxal bundle of non-intersecting type) and its symmetric G* with respect to line (f).
This system of circles determines a one parametric family of cyclic quadrilaterals whose diagonals pass through F. More precisely, for each point N of the arc (XY) of the circle (e), containing the point A, there is one and only one cyclic quadrilateral (q) inscribed in (c) with the properties:
1) The point N* is the other intersection point of (e) and the line NC. The diagonals of (q) intersect at F and have their middles at N, N* respectively.
2) The pairs of opposite sides of (q) intersect at the diametral points D, E of circle (d).
3) The centroids of (q), coincide with the middles of NN* and lie on the arc (XHY) of the circle with diameter CH.
I call (d) the [Orthocycle] of the cyclic quadrilateral. It seems to be connected with many properties of the cyclic quadrilateral (q).

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All the properties follow directly from the discussion made in the file CyclicProjective.html .

Fixing the system of the two circles and the point (c,d, F) and varying only N, there is a particular place for N for which the corresponding quadrilateral q becomes [bicentric]. See the file Orthocycle2.html .

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