HYACINTHOS 25224

[Antreas P. Hatzipolakis]:

 

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

Denote:

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

The circumcircles of IA'Na, IB'Nb, IC'Nc are coaxial.

Second point of intersection?

[Peter Moses]:

Hi Antreas,

 

a (a^2-b^2+b c-c^2) (2 a^4-a^3 b-a^2 b^2+a b^3-b^4-a^3 c+2 a^2 b c-a b^2 c-a^2 c^2-a b c^2+2 b^2 c^2+a c^3-c^4)::
on lines {{1,104},{3,2817},{10,6718},{ 36,186},{40,10696},{77,102},{ 117,515},{124,1125},{151,5731} ,{214,3738},{651,6326},{993, 1060},{999,1486},{1068,4299},{ 1104,3756},{1319,1361},{1364, 2646},{1385,2818},{1394,6261}, {1565,2792},{1699,10732},{ 1718,10090},{1807,2801},{2716, 7012},{2816,3184},{3315,3333}, {4296,11012},{4347,11249},{ 5886,10747},{6711,10165}}.

Search 0. 925617333318200697170623418441 .
Midpoint of X(i) and X(j) for these {i,j}: {{1, 109}, {40, 10696}}.

Reflection of X(i) in X(j) for these {i,j}: {{10, 6718}, {124, 1125}}.

X[102] - 3 X[3576], X[151] + 3 X[5731], 2 X[6711] - 3 X[10165], 3 X[1] - X[10703], 3 X[109] + X[10703], 3 X[1699] - X[10732], 3 X[5886] - X[10747].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,603,5884).
isoconjugate of X (j) and X (j) for these (i,j): {{80,102},{655,2432}}.
X (934)-Ceva conjugate of X (3960).
crosspoint of X(4585) & X(7045)
crossdifference of every pair of points on line X(2161) X(2432).
inverse in the incircle of X(11570).
barycentric product X (i) X (j) for these {i,j}: {{320,2182},{515,3218},{2406, 3738}}.
barycentric quotient X (i)/X(j) for these {i,j}: {{1455,2006},{2182,80},{2425, 2222},{3738,2399},{7113,102},{ 8648,2432}}.

 

Best regards,
Peter Moses.