HYACINTHOS 26086

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and P, Q two isogonal conjugate points..

Denote:

A', B', C' = the midpoints of AP, BP, CP, resp.

A", B", C" = the reflections of Q in BC, CA, AB, resp.

1. P = Q = I

 

2. P = H, Q = O

The circumcircles of PA'A", PB'B", PC'C" are coaxial.

 

2nd intersections ?

 

 

[Peter Moses]:

Hi Antreas,

 

1).

X(1155).

 

2).

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

on lines {{3,625},{4,69},{5,187},{30, 114},{182,7841},{297,5972},{ 381,2080},{524,9880},{538, 6321},{542,8352},{547,5215},{ 575,7790},{598,8590},{631, 7910},{842,10296},{1353,1570}, {1478,5194},{1479,5148},{1691, 10358},{1692,7745},{2021,5475} ,{2031,3767},{2076,10356},{ 2459,10577},{2460,10576},{ 3095,7843},{3398,7861},{3564, 5107},{3843,9301},{5025,13335} ,{5097,7812},{5104,10516},{ 5184,5587},{5613,8594},{5617, 8595},{5999,10722},{6033, 10242},{6054,8597},{6390, 10992},{6655,13334},{7574, 10748},{7773,9737},{7799, 13172},{7809,10723},{7817, 11842},{7918,10359},{9775, 10989},{10104,10631},{11178, 11317},{11303,13349},{11304, 13350},{11645,12177}}.

Midpoint of X(i) and X(j) for these {i,j}: {{4, 316}, {842, 10296}, {5999, 10722}, {6054, 8597}}.

Reflection of X(i) in X(j) for these {i,j}: {{3, 625}, {187, 5}, {10992, 6390}}.

2 X[1353] - 3 X[1570], 3 X[381] - X[2080], 4 X[547] - 3 X[5215], X[5184] - 3 X[5587], 5 X[3843] - X[9301], X[6033] + 3 X[10242], X[5104] - 3 X[10516], 3 X[7809] + X[10723], 3 X[6787] - X[11674], 3 X[7799] - X[13172].

inverse in Polar circle of X(6403).

Searches: {2. 94662722086188142533114485636, 0. 030848462036419080271953257070 4, 2. 25932606009983600508493676005} .

 

Best regards,

Peter Moses.