[1] The intersection points of pairs of sides C*=(AB,A'B'), A*=(BC,B'C'), B*=(CA,C'A') are collinear. The three points are on the

[2] C* is the

[3] The three circles a', b', c' belong to the same circle bundle (I) which is orthogonal to the bundle (II) generated by the circumcircle and the Euler circle.

[4] In the case of accute-angled triangle the circle bundle (I) is of intersecting type. The two common bundle points H', H'' are on the Euler line and the centroid G is harmonic conjugate to the orthocenter H with respect to H', H''.

[5] The

[6] h is the polar with respect to the circumcircle of the triangle center X(25).

[1] C*, B*, A* define the

Besides the circles c', b', a' are orthogonal to both Euler and Circumcircle, hence they belong to the bundle (II) of circles simultaneously orthogonal to Euler and Circumcircle. This proves the orthogonality of h to the Euler line.

[2], [3] Are consequences of the previous arguments.

[4] For accute-angled triangles the Euler and the circum-circle do not intersect, hence the statement about the bundle (I). For the rest see the exercise on circle-bundles in CircleBundleMember.html .

[5] and [6] involve calculations with trilinears as defined in the reference given below for

Orthic.html

Orthocenter.html

Orthocenter2.html

Kimberling, Clark.

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