[alogo] Orthic axis

Consider a triangle ABC and its altitudes AA', BB', CC' defining with their intersections the orhtocenter H and through the intersections with the sides of the triangle the orthic triangle A'B'C'.
[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 trilinear polar h of the orthocenter. h coincides witht the radical axis of the circumcircle and Euler circle and is orthogonal to the Euler line of the triangle.
[2] C* is the radical center of the circumcircle, Euler circle and circle with diameter AB. There is a circle c' centered at C* simulatenously orthogonal to the previously referred three circles. The intersection point C'' of c' and the circle with diameter AB is on the altitude CC'. Analogous properties hold for the other points B* and A* and the analogously defined points A'', B'' and circles a', b'.
[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 orthocentroidal circle with diameter HG belongs to bundle (II) and is centered at X(381).
[6] h is the polar with respect to the circumcircle of the triangle center X(25).

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]
[2_0] [2_1] [2_2] [2_3]
[3_0] [3_1] [3_2] [3_3]

[1] C*, B*, A* define the trilinear polar of the orthocenter. From their definitions these points are radical centers of triples of circles. f.e. C* is intersection point of the radical axes of circle pairs (Euler, (AB)), (circumcircle, (AB)), where (AB) denotes the circle with diameter AB. Hence the radical axis of (Euler, Circumcircle) passes also through C*. Analogous properties hold also for B* and A* and this identifies h with the radical axis of (Euler, Circumcircle).
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 triangle centers.

See Also



Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed. rev. enl. . Dublin, Hodges, Figgis, & Co., 1893, p. 436.
Kimberling, Clark. Encyclopedia of Triangle Centers http://faculty.evansville.edu/ck6/encyclopedia/ETC.html

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