HYACINTHOS 25216

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and IaIbIc the antipedal triangle of I.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

N1, N2, N3 = the NPC centers of IaBC, IbCA, IcAB, resp

 

The circumcircles of IN1Na, IN2Nb, IN3Nc are coaxial.

Second point of intersection?

[Peter Moses]:

Hi Antreas,

 

a^5 b^2-a^4 b^3-2 a^3 b^4+2 a^2 b^5+a b^6-b^7+4 a^5 b c-5 a^4 b^2 c+a^3 b^3 c+4 a^2 b^4 c-5 a b^5 c+b^6 c+a^5 c^2-5 a^4 b c^2+8 a^3 b^2 c^2-6 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2-a^4 c^3+a^3 b c^3-6 a^2 b^2 c^3+10 a b^3 c^3-3 b^4 c^3-2 a^3 c^4+4 a^2 b c^4-a b^2 c^4-3 b^3 c^4+2 a^2 c^5-5 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7::
on lines {{1,5},{3,153},{10,2771},{30, 100},{104,140},{149,381},{214, 1329},{226,6797},{390,6929},{ 528,3845},{546,10738},{549, 993},{550,2829},{632,6713},{ 1145,3869},{1532,5844},{1596, 1862},{2800,3678},{2801,3826}, {2932,5552},{2950,6259},{3627, 5840},{3853,10724},{4293,6924} ,{5066,10707},{5218,6914},{ 5657,9809},{5818,9803},{6174, 8703},{6224,11681},{6246,7680} ,{6945,10247},{9956,10265}}.

Search -3. 62765501758569357055442828260.

Midpoint of X(i) and X(j) for these {i,j}: {{3, 153}, {100, 10742}, {355, 6326}, {2950, 6259}}.

Reflection of X(i) in X(j) for these {i,j}: {{5, 119}, {104, 140}, {1484, 5}, {8703, 6174}, {10265, 9956}, {10707, 5066}, {10724, 3853}, {10738, 546}}.

3 X[5] - 2 X[11], X[11] - 3 X[119], X[149] - 3 X[381], 4 X[11] - 3 X[1484], 4 X[119] - X[1484], 3 X[549] - 4 X[3035], X[5531] + 3 X[5587], 3 X[5886] - X[6264], 3 X[5660] - X[6265], 5 X[632] - 4 X[6713], 9 X[5660] - X[7972], 3 X[6265] - X[7972], X[7993] - 5 X[8227], 5 X[5818] - X[9803], 3 X[5657] + X[9809], 3 X[355] - X[9897], 3 X[6326] + X[9897], X[100] + 3 X[10711], 9 X[10711] - X[10728], 3 X[100] + X[10728], X[10728] - 3 X[10742], 3 X[10711] - X[10742].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (100,10711,10742),(7951,9897, 11).

on K682.

 

Best regards,

Peter Moses.