Σάββατο, 6 Απριλίου 2013

ORTHOLOGIC, EULER LINE

Let ABC be a triangle, A'B'C', A"B"C" the medial, orthic triangles, resp. and A*,B*,C* points on AA",BB",CC", resp. such that: A*A/A*A" = B*B/B*B" = C*C/C*C" = t.

Denote:

Ab = (Parallel to BC through A*) /\ AC

Ac = (Parallel to BC through A*) /\ AB

Bc = (Parallel to CA through B*) /\ BA

Ba = (Parallel to CA through B*) /\ BC

Ca = (Parallel to AB through C*) /\ CB

Cb = (Parallel to AB through C*) /\ CA

1. Oa, Ob, Oc = the circumcenters of A'AbAc, B'BcBa, C'CaCb, resp.

The triangles ABC, OaObOc are orthologic

The locus of the orthologic center (OaObOc, ABC), as t varies, is the Euler line of ABC.

Which is the locus of the other orthologic center (ABC, OaObOc)?

2. Na, Nb, Nc = the NPCs centers of A'AbAc, B'BcBa, C'CaCb, resp.

The triangles ABC, NaNbNc are orthologic

The locus of the orthologic center (NaNbNc, ABC), as t varies, is the Euler line of ABC.

Which is the locus of the other orthologic center (ABC, NaNbNc)?

Antreas P. Hatzipolakis, 6 April 2013

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