HYACINTHOS 24721

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

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

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

NaNbNc, ABC are cyclologic.

With additional property: Circumcyclologic (*)

The circumcircles of ANbNc, BNcNa, CNaNb concur on the circumcircle of ABC

The circumcircles of NaBC, NbCA, NcAB concur on the circumcircle of NaNbNc.

(*) Hyacinthos #23137

 

[Peter Moses]:

Hi Antreas,

 

>The circumcircles of ANbNc, BNcNa, CNaNb concur on the circumcircle of ABC

X(930).

 

>The circumcircles of NaBC, NbCA, NcAB concur on the circumcircle of NaNbNc.

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

on lines {{2,10095},{3,54},{5,3917},{ 20,5876},{26,1350},{30,1216},{ 49,6636},{51,632},{52,549},{ 140,143},{156,394},{185,8703}, {381,7999},{389,3530},{399, 8718},{547,10110},{550,5562},{ 568,3523},{631,5946},{1092, 7502},{1112,10018},{1147,3098} ,{1511,7488},{1656,7998},{ 2781,7555},{2889,3448},{3060, 3526},{3567,5054},{3627,5891}, {3628,3819},{3850,10170},{ 5070,9781},{5944,7512},{6030, 9705}}.

 

Anticomplement X[10095].

Complement X[10263].

Midpoint of X(i) and X(j) for these {i,j}: {{3, 6101}, {20, 5876}, {550, 5562}}.

Reflection of X(i) in X(j) for these {i,j}: {{140, 5447}, {143, 140}, {389, 3530}, {5446, 3628}, {10263, 10095}}.

 

X[52] - 3 X[549], 3 X[51] - 5 X[632], X[3] + 3 X[2979], 3 X[568] - 7 X[3523], 3 X[3060] - 7 X[3526], 2 X[3628] - 3 X[3819], X[5] - 3 X[3917], 5 X[3567] - 9 X[5054], 3 X[3819] - X[5446], X[143] - 4 X[5447], 3 X[143] - 4 X[5462], 3 X[140] - 2 X[5462], 3 X[5447] - X[5462], 5 X[3] - X[5889], 15 X[2979] + X[5889], 7 X[5889] - 15 X[5890], 7 X[3] - 3 X[5890], 7 X[2979] + X[5890], X[3627] - 3 X[5891], 3 X[1216] - X[5907], 5 X[631] - 3 X[5946], 3 X[2979] - X[6101], X[5889] + 5 X[6101], 3 X[5890] + 7 X[6101], 9 X[5890] - 7 X[6102], 3 X[5889] - 5 X[6102], 3 X[3] - X[6102], 9 X[2979] + X[6102], 3 X[6101] + X[6102], 5 X[631] - X[6243], 3 X[5946] - X[6243], 5 X[1656] - 9 X[7998], 3 X[381] - 7 X[7999], X[185] - 3 X[8703], 11 X[5070] - 7 X[9781], 3 X[547] - 2 X[10110], 2 X[3850] - 3 X[10170].

 

(J^2 - 8) X[3] - (J^2 - 4) X[54], where J = |OH|/R

 

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,10263,10095),(3,2979,6101), (631,6243,5946),(1147,3098, 7525),(3819,5446,3628).

 

Best regards,

Peter Moses.