HYACINTHOS 25470

[Antreas P. Hatzipolakis]:

Let ABC be a triangle, AhBhCh the pedal triangle of H and O' the Poncelet point of ABCO [ O' = X125]

(ie O' is the point the NPCs of ABC, OBC, OCA, OAB are concurrent at).

Denote:

hA = the second intersection of AhO' and the NPC of OBC [other than O']
hB = the second intersection of BhO' and the NPC of OCA
hC = the second intersection of ChO' and the NPC of OAB

O', hA, hB, hC are concyclic.

Which point is the center of the circle?

 

[Angel Montesdeoca]:


 The center of the circle   passing through  O',.b4, hA, hB, hC is W = 3 X(3) + X(155)
 
 W = ( a^2 (a^2-b^2-c^2) (2 a^6+2 a^2 b^2 c^2-3 a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)) : ... : ...),
 
  with (6-9-13)-search numbers (5.02069206557260, 3.50252376074738, -1.10140215195116).
 
  W is the midpoint of X(i) and X(j) for these {i,j}:  {3,1147}, {155,7689}, {156,11250}.
 
 W is the reflection of X(i) in X(j) for these {i,j}:  {5448,9820}, {5449,140}.
 
  W on lines: {2, 9927}, {3, 49}, {4, 11449}, {5, 1511}, {20, 5654}, {24, 5446},  {26, 11202}, {30, 5448}, {52, 186}, {54, 5504}, {68, 631}, {74, 9705}, {110, 3520}, {140, 5449}, {156, 6000}, {182, 8548}, {378, 10539}, {382, 1495}, {511, 1658}, {539, 549}, {541, 5894}, {550, 5944}, {567, 2931}, {569, 5892}, {578, 5462}, {1069, 5217}, {1152, 8909}, {1614, 2071}, {3043, 11562}, {3157, 5204}, {3523, 6193}, {3524, 11411}, {3530, 3564}, {3576, 9928}, {3855, 10546}, {5010, 6238}, {5646, 7393}, {5657, 9933}, {5663, 10226}, {5890, 9545}, {6146, 10257}, {6200, 10666}, {6241, 9544}, {6396, 10665}, {6418,8912}, {6642, 11425}, {6689, 7399}, {6699, 10116}, {7280, 7352}, {7488, 10625}, {7503, 10170}, {7506, 11424}, {7514, 9938}, {7526, 9306}, {7575, 10263}, {8546, 8681}, {9707, 11413}, {10020, 10182},  {10298, 11412}, {10540, 11381}, {10645, 10662}, {10646, 10661}.

Angel Montesdeoca