There are some remarkable analogies between this and the usual Miquel configuration, resulting by taking three arbitrary points on the sides of the triangle. A discussion on this topic is contained in Miquel_Point.html . In the present configuration hold the following properties.

[1] Let {A',B',C'} be arbitrary points on line-sides respectively {BC,CA,AB} and {A

[2] Every other tripple of lines {AB

[3] Triangles A

The clue for all these properties lies in the following two relations of angles.

First, considering the intersection O of circumcircles (BA

Second, angle(C'CA

[1] Consider the four triangles {ABC,DAB,DBC,DCA} defined by a basic triangle ABC and an additional point D.

The Euler circles of these four triangles intersect at a point P.

[2] The unique rectangular parabola passing through four non-orthocentroidal points {A,B,C,D} has its center at P.

The statement on the four Euler circles is proved in EulerCirclesFour.html . The other assertion follows from the general properties of triangles inscribed in rectangular hyperbolas (see RectHypeCircumscribed.html ). Any triangle inscribed in a rectangular hyperbola has its center on its Euler circle. Hence in our case P is the center of the hyperbola.

Miquel_Point.html

RectHypeCircumscribed.html

Similarity.html

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