HYACINTHOS 24098

Antreas P. Hatzipolakis

 

[APH]

 

Let ABC be a triangle and A'B'C' the pedal triangle O.

Denote:

A",B",C" = the orthogonal projections of O on B'C', C'A', A'B', resp.

Ab = the intersection of OA" and AB'

Ac = the intersection of OA" and AC'

(Nab), (Nac) = the NPCs of A"B'Ab, A"C'Ac, resp.

(Nbc), (Nba) = the NPCs of B"C'Bc, B"A'Ba, resp.

(Nca), (Ncb) = the NPCs of C"A'Ca, C"B'Cb, resp.

S1 = the radical axis of (Nba), (Nca).

S2 = the radical axis of (Ncb), (Nab)

S3 = the radical axis of (Nac), (Nbc)

 

T1 = the radical axis of (Nbc), (Ncb).

T2 = the radical axis of (Nca), (Nac)

T3 = the radical axis of (Nab), (Nba)

 

1. S1, S, S3 are concurrent (on the Euler line?)
2. T1, T2, T3 are concurrent (on the Euler line)
3. Naturally we can ask for the loci (P instead of O), but I guess they are complicated

APH

[Peter Moses]:

Hi Antreas,

 

>1. S1, S2, S3 are concurrent (on the Euler line?)

X(6676)

 

2. Not concurrent.

 

Best regards,

Peter Moses