Orthocycle2

Orthocycle and bicentric

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. This family is studied in Orthocycle.html . There is a unique member of this family, q = RSTU, which is a bicentric quadrilateral. This is defined by the intersection point Q of the circle (e) with the line CG. The following properties are valid:
1) Q, Q* are the middles of the diagonals of q.
2) The bisectors of angles SDR and SET pass through G, which is the center of the inscribed circle g(G,r), with r^2 = |GF||GI|.
3) For all the varying (d) of the coaxal bundle II the corresponding bicentrics q have the same incircle (g).

All the properties follow directly from the discussion made in the reference given and the remarks in Bicentric.html . In particular, the last statement on the incircle, follows from the independence of r from the particular (d). Thus, fixing the circumcircle (c) and F, all the bicentrics with diagonals through F are parameterized 1-1 by the member-circles (d) of the circle bundle II. All of them having the same incircle g(G,r).

From all these bicentrics with common incircle (g) there are exactly two that have the diagonals orthogonal. Indeed QQ* is then simultaneously diameter of (e) and passes through the center C of (d). This is only possible when Q coincides with A, or QQ* is a diameter of (e) orthogonal to AF. In both cases the bicentric q is then symmetric with respect to the line IA. Thus, every bicentric and simultaneously orthodiagonal is axis-symmetric. The corresponding orthocycle (d) is then the minimal member of the circle bundle II (the one with diameter GG*) or the maximal member of that family, which is the line GG*. The first case gives the shape of a kite with diagonals parallel to lines GG* and (f). The second case gives an isosceles trapezium with orthogonal diagonals and two sides parallel to line (f). See the file Orthodiagonal2.html for a discussion on orthodiagonals.

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