HYACINTHOS 25824

 [Antreas P. Hatzipolakis]:

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

Denote:

Aa, Ab, Ac = the orthogonal projections of A' on OA, OB, OC, resp.

Ba, Bb, Bc = the orthogonal projections of B' on OA, OB, OC, resp.

Ca, Cb, Cc = the orthogonal projections of C' on OA, OB, OC, resp.

Ra, Rb, Rc = the Euler lines of AaAbAc, BaBbBc, CaCbCc, resp.

 

1. Ra, Rb, Rc are concurrent. Point?

 

2. The parallels to Ra, Rb, Rc through A, B, C, resp. are concurrent on the circumcircle at X1141

(Variation: Hyacinthos #24187)

 

 

[Peter Moses]:

 

 

Hi Antreas,

 

1) 2 a^12-8 a^10 b^2+14 a^8 b^4-15 a^6 b^6+11 a^4 b^8-5 a^2 b^10+b^12-8 a^10 c^2+16 a^8 b^2 c^2-9 a^6 b^4 c^2-4 a^4 b^6 c^2+8 a^2 b^8 c^2-3 b^10 c^2+14 a^8 c^4-9 a^6 b^2 c^4+4 a^4 b^4 c^4-3 a^2 b^6 c^4+3 b^8 c^4-15 a^6 c^6-4 a^4 b^2 c^6-3 a^2 b^4 c^6-2 b^6 c^6+11 a^4 c^8+8 a^2 b^2 c^8+3 b^4 c^8-5 a^2 c^10-3 b^2 c^10+c^12:: 

on lines {{2,137},{3,128},{140,6592},{ 631,1141},{632,1263},...}.

Complement X[137]. 

Midpoint of X(i) and X(j) for these {i,j}: {{3, 128}, {137, 930}, {140, 6592}}.

3 X[2] + X[930], 5 X[631] - X[1141], 5 X[632] - X[1263], 9 X[2] - X[11671], 3 X[137] - X[11671], 3 X[930] + X[11671], 3 X[140] - X[12026], 3 X[6592] + X[12026].

{X(2),X(930)}-harmonic conjugate of X(137).

on the NP circle of the medial triangle.

 

 

Best regards,

Peter Moses.