HYACINTHOS 26141

[Antreas P. Hatzipolakis]:

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

Denote:

Ab, Ac = the reflections of B', C' in AO, resp.

Na = the NPC center of OAbAc. Similarly Nb, Nc.

ABC, NaNbNc are orthologic
(and A'B'C', NaNbNc are orthologic)

 

[Peter Moses]:

 

 

Hi Antreas,

 

(ABC,  NaNbNc) : X(265).

(A'B'C',  NaNbNc) : X(1511).

 

(NaNbNc, ABC) & (NaNbNc, A'B'C') :

2 a^10-4 a^8 b^2+a^6 b^4+a^4 b^6+a^2 b^8-b^10-4 a^8 c^2+2 a^6 b^2 c^2+2 a^4 b^4 c^2-3 a^2 b^6 c^2+3 b^8 c^2+a^6 c^4+2 a^4 b^2 c^4+4 a^2 b^4 c^4-2 b^6 c^4+a^4 c^6-3 a^2 b^2 c^6-2 b^4 c^6+a^2 c^8+3 b^2 c^8-c^10:: 

on lines {{3,12278},{4,13353},{5, 1495},{20,3581},{30,143},{54,7 574},{265,7512},{1154,6146},{1 594,10610},{2072,5944},{3521,1 0296},{3575,12006},{3850,13419 },{5073,9777},{5663,12605},{ 6102,12225},{6689,13413},{ 6756,13364},{7525,9927},{ 10095,11819},{10263,12022},{ 11591,12362},{12370,13391}}.

midpoint of X(i) and X(j) for these {i,j}: {{5, 11750}, {6102, 12225}}.

reflection of X(i) in X(j) for these {i,j}: {{3575, 12006}, {6146, 11565}, {11264, 6146}, {11591, 12362}, {11819, 10095}, {13419, 3850}}.

Searches: {3.642065729313682552780477890 97,3.6525164917282683891939423 9028,-0.5689541951261217291453 63271216}}.

 

Best regards,

Peter Moses.