## Πέμπτη, 10 Απριλίου 2014

Let ABC be a triangle, P a point and A'B'C' the cevian triangle of P. Denote:

Ab, Ac = the reflections of A' in AB, AC. resp.

A2, A3 = the reflections of A' in BB', CC', resp.

Oab, Oac = the circumcenters of BAbA2, CAcA3, resp.

Similarly (cyclically):

Obc, Oba and Oca, Ocb.

1. P = O.

The radical axes R1 =:((Oab),(Oac)), R2 =:((Obc),(Oba)), R3 =:((Oca),(Ocb))are concurrent at O.

The reflections of R1,R2,R3 in BC,CA,AB are the lines AN,BN,CN, resp.

2. P = H.

The radical axes R1 =:((Oab),(Oac)), R2 =:((Obc),(Oba)), R3 =:((Oca),(Ocb)) are concurrent on the Euler line of ABC.

The reflections of the radical axes S1 =:((Obc), (Ocb)), S2 =:((Oca),(Oac)), S3 =: ((Oab), (Oba)) in BC,CA,AB, resp. are concurrent.

The reflections of the radical axes T1 =: ((Oba), (Oca)), T2 =:((Ocb),(Oab)), T3 =:((Oac), (Obc)) in Bc,CA,AB are concurrent.

The triangles: bounded by the lines (T1,T2,T3) and the orthic A'B'C' are paralle;ogic.

Antreas P. Hatzipolakis, 10 April 2014