CYCLOLOGIC

Let ABC be a triangle

Denote

1. Oa, Ob, Oc = the circumcenters of HBC, HCA, HAB, resp.

ABC, OaObOc are cyclologic, since Oa, Ob, Oc are the reflections of O in BC,CA,AB, resp.
Cyclologic center (OaObOc, ABC) = antigonal conjugate of O = X(265)

2. Sa, Sb, Sc = the X(54) of HBC, HCA, HAB, resp.

ABC,SaSbSc are cyclologic

Cyclologic centers?

ETC

X(72482) = X(2)X(9291)∩X(4)X(290)

Barycentrics    b^2*c^2*(a^4-a^2*b^2-a^2*c^2+2*b^2*c^2)*(-a^4+(b^2-c^2)^2)^2 : :

Antreas Hatzipolakis and Ercole Suppa, euclid 9604.

X(72482) lies on these lines: {2, 9291}, {4, 290}, {20, 16089}, {69, 57677}, {76, 1093}, {107, 37465}, {194, 47739}, {253, 264}, {276, 3090}, {311, 59139}, {317, 42355}, {381, 42368}, {393, 2998}, {683, 6524}, {1235, 52661}, {1975, 15143}, {2052, 2996}, {5071, 55079}, {6331, 6337}, {6530, 44144}, {8795, 15077}, {11185, 62274}, {13450, 44146}, {14618, 53173}, {16081, 64983}, {18817, 59428}, {22456, 53783}, {27376, 42359}, {40680, 68535}, {52448, 62949}

X(72482) = polar conjugate of X(51336)
X(72482) = lies on all circumconics with perspector on the line {30476, 40887}
X(72482) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(4),X(1968)}, {A,B,C,X(69),X(59527)}, {A,B,C,X(253),X(290)}, {A,B,C,X(263),X(40951)}, {A,B,C,X(6526),X(57677)}, {A,B,C,X(8798),X(9306)}, {A,B,C,X(37174),X(37199)}, {A,B,C,X(52581),X(60199)}, {A,B,C,X(53173),X(62542)}, {A,B,C,X(57843),X(60841)}}
X(72482) = pole of tripolar of X(51336) with respect to polar circle
X(72482) = pole of the line {16229, 62521} with respect to Steiner circumellipse
X(72482) = pole of the line {417, 10607} with respect to Wallace hyperbola
X(72482) = trilinear pole of line {X(30476), X(40887)}
X(72482) = barycentric product X(i)*X(j) for these (i, j): {264, 9308}, {290, 40887}, {823, 17893}, {1957, 1969}, {1958, 57806}, {1968, 18022}, {1975, 2052}, {6331, 16229}, {6528, 30476}, {9306, 18027}, {15143, 60199}, {17478, 57973}
X(72482) = barycentric quotient X(i)/X(j) for these (i, j): {4, 51336}, {92, 9255}, {158, 9258}, {264, 9289}, {393, 9292}, {1957, 48}, {1958, 255}, {1968, 184}, {1975, 394}, {2052, 9307}, {2451, 39201}, {2996, 60834}, {6528, 43188}, {9306, 577}, {9308, 3}, {15143, 3289}, {15352, 65837}, {16229, 647}, {17215, 4091}, {17478, 822}, {17893, 24018}, {22089, 32320}, {30476, 520}, {37199, 3167}, {40887, 511}, {59527, 6509}, {59561, 22401}, {60841, 60833}, {64983, 43727}
X(72482) = trilinear product X(i)*X(j) for these (i, j): {92, 9308}, {107, 17893}, {158, 1975}, {264, 1957}, {811, 16229}, {821, 59527}, {823, 30476}, {1821, 40887}, {1958, 2052}, {1968, 1969}, {2451, 57973}, {6528, 17478}, {9306, 57806}
X(72482) = trilinear quotient X(i)/X(j) for these (i, j): {48, 9308}, {92, 51336}, {158, 9292}, {184, 1957}, {255, 1975}, {264, 9255}, {520, 17893}, {577, 1958}, {810, 16229}, {820, 59527}, {822, 30476}, {1755, 40887}, {1968, 9247}, {1969, 9289}, {2052, 9258}, {9306, 52430}, {9307, 57806}, {17215, 23224}, {17478, 39201}, {43188, 57973}
X(72482) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6528, 18027, 4}, {9291, 62576, 2}

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X(72483) = X(290)X(15740)∩X(311)X(36889)

Barycentrics    b^2*c^2*(a^4-a^2*b^2-a^2*c^2+4*b^2*c^2)*(a^4+6*a^2*b^2+b^4-2*a^2*c^2-2*b^2*c^2+c^4)*(a^4-2*a^2*b^2+b^4+6*a^2*c^2-2*b^2*c^2+c^4) : :

Antreas Hatzipolakis and Ercole Suppa, euclid 9604.

X(72483) lies on the circumconic {A,B,C,X(290),X(5485)} and these lines: {290, 15740}, {311, 36889}, {5485, 37874}, {9462, 52223}

X(72483) = barycentric product X(i)*X(j) for these (i, j): {37874, 69380}
X(72483) = barycentric quotient X(i)/X(j) for these (i, j): {5651, 5065}, {69380, 17811}
X(72483) = trilinear quotient X(i)/X(j) for these (i, j): {1496, 69380}

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G - Orthologic

Let ABC be a triangle, P = G = X(2) and Q a point on the Euler line.

Denote:

Bc, Cb = the orthogonal projections of B, C on GC, GB, resp.

Qa = same to Q point of the triangle ABcCb.
Similarly Qb, Qc.

ABC, QaQbQc are orthologic.

For Q = G = X(2)
Orthologic center (ABC, QaQbQc) = G* = ?
Orthologic center (QaQbQc, ABC) = G** = ?

For Q = X(3) = O:
Orthologic center (ABC, QaQbQc) = O* = X(36889)
Orthologic center (QaQbQc, ABC) = O** = X(1352)
Euclid 9541

Q = H = X(4)
Orthologic center (ABC, QaQbQc) = H* = X(3)= O
Orthologic center (QaQbQc, ABC) = H** = ?

Q = N = X(5)
Orthologic center (ABC, QaQbqc) = N* = ?
Orthologic center (QaQbQc, ABC) = N** = ?

The locus of the orthologic center (QaQbQc, ABC) = Q**, as Q moves on the Euler line, is a line.
(OQ/OH = O**Q**/O**H**)

Locus of the orthologic center (ABC, QaQbQc) ?

H - Orthologic

Let ABC be a triangle, P = H = X(4) and Q a point on the Euler line.

Denote:

Bc, Cb = the orthogonal projections of B, C on HC, HB, resp.

Qa = same to Q point of the triangle ABcCb.
Similarly Qb, Qc.

ABC, QaQbQc are orthologic.

For Q = G = X(2)
Orthologic center (ABC, QaQbQc) = G* = ?
Orthologic center (QaQbQc, ABC) = G** = G of orthic = X(51)

For Q = X(3) = O:
Orthologic centers = X(4) = H

Q = H = X(4)
Orthologic center (ABC, QaQbQc) = H* = X(3) = O
Orthologic center (QaQbQc, ABC) = H** = ?

For Q = N = X(5)
Orthologic center (ABC, QaQbQc) = N* = ?
Orthologic center (QaQbQc, ABC) = N** = ?

The locus of the orthologic center (QaQbQc, ABC) = Q**, as Q moves on the Euler line, is a line. (The line {4,51})
(OQ/OH = O**Q**/O**H**)

Locus of the orthologic center (ABC, QaQbQc) ?

O - Orthologic

Let ABC be a triangle, P = O = X(3) and Q a point on the Euler line.

Denote:

Bc, Cb = the orthogonal projections of B, C on OC, OB, resp.

Qa = same to Q point of the triangle ABcCb.
Similarly Qb, Qc.

ABC, QaQbQc are orthologic.

Orthologic center (QaQbQc, ABC) = Q

For Q = G = X(2)
Orthologic center (ABC, QaQbQc) = G* = ?

For Q = X(3) = O:
Orthologic center (ABC, QaQbQc) = O* = X(72422) = X(2)X(9291)∩X(4)X(290)

For Q = H = X(4)
Orthologic center (ABC, QaQbQc) = H* = ?

For Q = N = X(5)
Orthologic center (ABC, QaQbQc) = N* = ?

Locus:
The locus of the orthologic center (ABC, QaQbQc) = Q*, as Q moves on the Euler line, is a CIRCLE

LOCI

Let ABC be a triangle and P a point.

Denote:
Bc, Cb = the orthogonal projections of B, C on PC, PB, resp.

A' = the other than A intersection the circumcircles of ABC and ABcCb
Similarly B',C'

La, B, Lc = Euler lines of A'BC, B'CA, C'AB, resp.

1. Which is the locus of P such that ABC, A'B'C' are orthologic?
O lies on the locus
Orthologic center (ABC, A'B'C') = (3) = O
Orthologic center ( A'B'C', ABC) = Χ(20)

2. Which is the locus of P such that the parallels to La,Lb, Lc through A, B, C,resp, are concurrent?
O lies on the locus.
.

Circumcenters - Orthologic

Let ABC be a triangle and P a point.

Denote:

Bc, Cb = the orthogonal projections of B, C on PC, PB, resp.

Oa = the circumcenter of ABcCb.
Similarly Ob, Oc.

Which is the locus of P such that ABC, OaObOc are orthologic?

H, O, G lie on the locus.