(1) From the discussion in the previous cited reference and the one in EulerCircleProperty.html we know that each point C determines an angle(CBA) and the direction of one asymptotic line coincides with the bisector A

(2) To determine the hyperbola take the projections D, F of E on the asymptotics and determine the vertex G of the hyperbola, lying on the bisector of angle(DCF) and having CG

The small diagram shows (part of) the periodic graph of the function of length of y=|CG| (axis of the hyperbola) as a function of the polar angle of the asymptotic line. The lower peaks correspond to y=0, and are taken when C is at the feet of the altitudes, where the hyperbolas degenerate to two orthogonal lines. The upper local maxima correspond to the positions of C, for which the fourth intersection point I of the circumcircle, becomes identical with a vertex of the triangle. Then the hyperbola is tangent at this vertex to the circumcircle.

Take the symmetrics {D,G,...} of the vertices of the triangle with respect to I and draw the general conic passing through five out of the six created points.

The figure shows also the locus of perspectors of the circumscribed rectangular hyperbolas which is the orthic axis of the triangle.

Triangle EFG is a

Bicentric2.html

EulerCircleProperty.html

FourEulerCircles.html

HyperbolaRectangular.html

Jerabek.html

Kiepert.html

OrthoRectangular.html

RectangularAsProp.html

RectHyperbola.html

RectHypeThroughFourPts.html

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