Σάββατο, 30 Μαρτίου 2013

ORTHOPOLAR CIRCLES [triangle]

Let ABC be a triangle, P, Q two points and O, Q1,Q2,Q3 the circumcenters of ABC, QBC, QCA, QAB, resp.

Denote:

P0 = the orthopole of PO wrt ABC

P1 = the orthopole of PQ1 wrt QBC

P2 = the orthopole of PQ2 wrt QCA

P3 = the orthopole of PQ3 wrt QAB

We have:

1. P0, P1, P2, P3 lie on the NPCs (N),(N1),(N2),(N3) of ABC, QBC, QCA, QAB, resp. (since the respective lines pass through the circumcenters of the respective triangles)

2. The NPCs of ABC, QBC, QCA, QAB concur at the Poncelet point Q* of Q wrt ABC.

CONJECTURE:

The points P0, P1, P2, P3, Q* are concyclic.

Antreas P. Hatzipolakis, 30 March 2013.

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