SIX ORTHOCENTERS - A CONIC

Let ABC be a triangle and A'B'C', A"B"C" the cevian triangles of H,G, resp. (orthic, medial tr.).

Denote: H1,H2,H3 = the orthocenters of A'B"C", B'C"A", C'A"B", resp.

h1,h2,h3 = the orthocenters of A"B'C', B"C'A', C"A'B', resp.

H1,H2,H3, H are concyclic. The center of the circle is the common circumcenter of A'B'C' and A"B"C", the N of ABC.

1. The triangles H1H2H3, h1h2h3 are perspective.

The six orthocenters lie on a conic (rectangular hyperbola) with center P, the perspector of the triangles.

2. Denote: O1,O2,O3 = the circumcenters of H1h2h3, H2h3h1, H3h1h2, resp.

The triangles A'B'C', O1O2O3 are perspective.

3. The NPC centers of the triangles H1h2h3, H2h3h1, H3h1h2, H1H2H3, h1h2h3 concur at P (center of the hyperbola)

Antreas P. Hatzipolakis, 12 May 2013

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1. The triangles H1H2H3, h1h2h3 are perspective. The six orthocenters lie on a conic (rectangular hyperbola) with center P, the perspector of the triangles.

P=X(389), CENTER OF THE TAYLOR CIRCLE.

2. Denote: O1,O2,O3 = the circumcenters of H1h2h3, H2h3h1, H3h1h2, resp.

The triangles A'B'C', O1O2O3 are NOT perspective.

Another case:

Denote: o1,o2,o3 = the circumcenters of h1H2H3, h2H3H1, h3H1H2, resp.

The triangles A"B"C", o1o2o3 are perspective, witn perspector X(52)= ORTHOCENTER OF ORTHIC TRIANGLE.

Angel Montesdeoca, 13 May 2013