SIX NPCs [N of medial, G of orthic]

Let ABC be a triangle and A'B'C' the cevian triangle of P.

Denote:

Nab = The NPC center of AA'B'

Nac = The NPC center of AA'C'

Nbc = The NPC center of BB'C'

Nba = The NPC center of BB'A'

Nca = The NPC center of CC'A'

Ncb = The NPC center of CC'B'

1. P = G (A'B'C' = medial triangle)

The lines NbaNca, NcbNab, NacNbc are concurrent at the NPC center of A'B'C'.

2. P = H (A'B'C' = orthic triangle)

2.1.The triangles ABC and bounded by the lines (NbcNcb, NcaNac, NabNba) are homothetic

The homothetic center is G'(centroid) of A'B'C'

2.2. The homothetic center of the medial triangle and (NbcNcb, NcaNac, NabNba) is the infinite point of the line GG'.

Antreas P. Hatzipolakis, 27 March 2013

2.3. The radical axes of ((Nab),(Nac)), ((Nbc),(Nba)),((Nca),(Ncb)) are concurrent.

Antreas P. Hatzipolakis, 1 April 2013