HYACINTHOS 25171

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

A'B'C' = the pedal triangle of I

A"B"C" = the reflection triangle of A'B'C'

(ie A",B",C" = the reflections of A',B',C' in B'C',C'A',A'B', resp.)

A*,B*,C* = the reflections of I in BC, CA, AB, resp.

A*B*C*, A"B"C" are cyclologic.

Cyclologic centers ?

[cyclologic center (A*B*C*, A"B"C") = the point of concurrence of the circumcircles of A*B"C",B*C"A", C*A"B"

cyclologic center (A"B"C", A*B*C*)  = the point of concurrence of the circumcircles of A"B*C*, B"C*A*, C"A*B*)

 

[Peter Moses]:

Hi Antreas,

 

cyclologic center (A*B*C*, A"B"C") =X(80).

cyclologic center (A"B"C, A*B*C*)  = a (a^7 b+a^6 b^2-3 a^5 b^3-3 a^4 b^4+3 a^3 b^5+3 a^2 b^6-a b^7-b^8+a^7 c-2 a^6 b c+a^5 b^2 c+2 a^4 b^3 c-a^3 b^4 c-2 a^2 b^5 c-a b^6 c+2 b^7 c+a^6 c^2+a^5 b c^2+2 a^4 b^2 c^2-a^3 b^3 c^2-2 a^2 b^4 c^2-a b^5 c^2-3 a^5 c^3+2 a^4 b c^3-a^3 b^2 c^3+2 a^2 b^3 c^3+3 a b^4 c^3-2 b^5 c^3-3 a^4 c^4-a^3 b c^4-2 a^2 b^2 c^4+3 a b^3 c^4+2 b^4 c^4+3 a^3 c^5-2 a^2 b c^5-a b^2 c^5-2 b^3 c^5+3 a^2 c^6-a b c^6-a c^7+2 b c^7-c^8)::
= 3 X[354] - 2 X[3024]
on lines {{1,10620},{46,399},{55,9904}, {65,5663},{74,2646},{110,1155} ,{146,1837},{354,3024},...}.

 

Best regards,

Peter Moses.

[Antreas P. Hatzipolakis]:

 

Thanks, Peter !!

Circumcenter Version:

Let ABC be a triangle.

Denote:

A'B'C' = the reflection triangle

(ie A',B',C' = the reflections of A,B,C in BC, CA, AB, resp.)

A*,B*,C* = the reflections of O in A, B, C, resp.

A*B*C*, A'B'C' are cyclologic.

Cyclologic centers ?

1. cyclologic center (A*B*C*, A'B'C') = the point of concurrence of the circumcircles of A*B'C',B*C'A', C*A'B'

2. cyclologic center (A'B'C', A*B*C*)  = the point of concurrence of the circumcircles of A'B*C*, B'C*A*, C'A*B*

1 = X80 of the tangential triangle = ETC point ?

 

[Peter Moses]:

Hi Antreas,

 

1) = X80 of the tangential triangle = ETC point ?

X(399).

 

2) a^12-4 a^10 b^2+7 a^8 b^4-6 a^6 b^6+a^4 b^8+2 a^2 b^10-b^12-4 a^10 c^2+8 a^8 b^2 c^2-6 a^6 b^4 c^2+4 a^4 b^6 c^2-8 a^2 b^8 c^2+6 b^10 c^2+7 a^8 c^4-6 a^6 b^2 c^4-a^4 b^4 c^4+6 a^2 b^6 c^4-15 b^8 c^4-6 a^6 c^6+4 a^4 b^2 c^6+6 a^2 b^4 c^6+20 b^6 c^6+a^4 c^8-8 a^2 b^2 c^8-15 b^4 c^8+2 a^2 c^10+6 b^2 c^10-c^12::
on lines {{2,137},{3,1263},{20,1141},{ 128,3091},{146,382},{388,7159} ,{497,3327},...}.

Anticomplement X[930].

Reflection of X(i) in X(j) for these {i,j}: {{3, 1263}, {20, 1141}, {930, 137}}.

3 X[2] - 4 X[137], 4 X[128] - 5 X[3091].

{X(137),X(930)}-harmonic conjugate of X(2).

X(i)-anticomplementary conjugate of X(j) for these (i,j): {{1,1510},{1510,8},{1994,7192} ,{2616,6101},{2964,523},{2965, 4560},{3518,7253}}.

inverse in the Orthoptic Circle of the Steiner Circumellipe of X(5966).

on the circumcircle of the reflected triangle.

on the circumcircle of the anticomplementary triangle.

Search = -20. 3911006850849702917792601674.

 

Best regards,

Peter Moses.