HYACINTHOS 24100

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)

[...]

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

APH

[Peter Moses]:

Hi Antreas,

 

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

X(6676)

 

[...]

 

Best regards,

Peter.

[APH]:

I think the parallels to S1, S2, S3 through A, B, C, resp. concur on the Euler line.

(and the parallels through A', B', C' too)

[Peter Moses]:

Hi Antreas,

 

>I think the parallels to S1, S2, S3 through A, B, C, resp. concur on the Euler line.

X(378).

 

>(and the parallels through A', B', C' too)
(a^2-b^2-c^2) (a^6 b^2-a^4 b^4-a^2 b^6+b^8+a^6 c^2+6 a^4 b^2 c^2+a^2 b^4 c^2-4 b^6 c^2-a^4 c^4+a^2 b^2 c^4+6 b^4 c^4-a^2 c^6-4 b^2 c^6+c^8):: on lines {{2,3},{11,1060},{12,1062},{ 68,1181},{113,127},{115,131},{ 265,1176},{311,339},{394,5654} ,...}.

 

Best regards,

Peter Moses