## Παρασκευή, 10 Φεβρουαρίου 2012

### EL GRECO CIRCLE

Continued from X3542

Let N, N1, N2, N3 be the NPC centers of ABC, A2BC, B2CA, C2AB.

Are  N,N1,N2,N3 lying on a circle (with center on the HO line [Euler Line], and for a point P instead of O, on the HP line)?

APH, 10 February 2012

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It is true for P = O (see the proof in the comment).
I name the circle as EL CRECO circle in honor of the great Cretan - Spanish painter DOMINICOS THEOTOKOPOULOS known as EL GRECO

APH

#### 2 σχόλια:

1. Αυτό το σχόλιο αφαιρέθηκε από έναν διαχειριστή ιστολογίου.

2. Suppose, A3B3C3 is the circumcevian triangle of H wrt ABC and A4B4C4 be the antipodal triangle of ABC. A2B2C2 is the cevian triangle of the De-Longchamps point of ABC wrt A3B3C3, then if we can prove that A4B4C4 and A2B2C2 are perspective with the perspector lying on OH, then the result follows from picasso circles problem.

To prove it, suppose, A5B5C5 be the cirumcevian triangle of the De-longchamps point of ABC wrt A4B4C4. Clearly, A5B5C5 is the antipodal triangle of A3B3C3. Now if we use the generalised problem I posted in the proof of picasso circles problem earlier on triangle A3B3C3(and take R=De-Longchamps Point of ABC, Q= O), then we get the required result.