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(72671) = COMPLEMENT OF X(16230)

Barycentrics    (b^2-c^2)*(-a^2+b^2+c^2)^2*(2*a^4-a^2*b^2+b^4-a^2*c^2-2*b^2*c^2+c^4) : :
X(72671) = 3*X[2]-X[16230], X[3]+X[6334], 2*X[5]-X[68327], X[684]-3*X[14417], X[684]+3*X[65723], X[684]+X[68791], X[14417]+X[65723], 3*X[14417]+X[68791], 3*X[65723]-X[68791], X[3265]+X[47194], 5*X[631]-X[44427], X[868]+X[65766], X[879]+X[6333], 3*X[3268]+X[53345], X[4226]+X[65976], X[5489]+X[41077], X[6563]+X[68781], X[30735]+X[50553], 3*X[53383]+X[65871], X[35364]-2*X[65408], X[39201]+X[60597], 5*X[40336]-X[65977], X[41078]+X[62438], 3*X[44564]-2*X[45259], X[56370]+X[65772]

Antreas Hatzipolakis and Ercole Suppa, euclid 9745.

X(72671) lies on these lines: {2, 16230}, {3, 690}, {5, 68327}, {30, 44921}, {114, 2974}, {122, 125}, {216, 2491}, {520, 3265}, {523, 4885}, {526, 13416}, {631, 44427}, {647, 6368}, {804, 45261}, {826, 52584}, {868, 65766}, {879, 6333}, {1040, 53563}, {1368, 53567}, {2799, 6036}, {3268, 53345}, {3284, 68793}, {4226, 65976}, {5489, 41077}, {5972, 6132}, {6563, 68781}, {6823, 66498}, {8552, 15115}, {8651, 65694}, {9003, 32257}, {14295, 62698}, {15116, 60342}, {15366, 55132}, {15760, 39509}, {17974, 39473}, {30735, 50553}, {30789, 53383}, {35067, 47406}, {35364, 65408}, {39072, 65484}, {39201, 60597}, {40336, 65977}, {41078, 62438}, {44529, 47138}, {44564, 45259}, {47216, 62502}, {47570, 55131}, {55267, 71409}, {56370, 65772}

X(72671) = midpoint of X(i) and X(j) for these {i,j}: {3, 6334}, {684, 68791}, {868, 65766}, {879, 6333}, {3265, 47194}, {4226, 65976}, {5489, 41077}, {6563, 68781}, {14417, 65723}, {30735, 50553}, {39201, 60597}, {41078, 62438}, {56370, 65772}
X(72671) = reflection of X(i) in X(j) for these {i,j}: {35364, 65408}, {68327, 5}
X(72671) = complement of X(16230)
X(72671) = perspector of the circumconic through X(68)and X(525)
X(72671) = center of the circumconic through X(868)and X(879)
X(72671) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(114),X(684)}, {A,B,C,X(125),X(3265)}, {A,B,C,X(230),X(68791)}, {A,B,C,X(520),X(20975)}, {A,B,C,X(525),X(66264)}, {A,B,C,X(647),X(57154)}, {A,B,C,X(1650),X(4226)}, {A,B,C,X(2974),X(6394)}, {A,B,C,X(3564),X(9033)}, {A,B,C,X(4131),X(18210)}}
X(72671) = center of the inconic with perspector X(17932)
X(72671) = lies on the inconics with perspectors X(n) for these n: {4226, 62645}
X(72671) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {2, 47097, 67641}, {3, 127, 131}, {4, 74, 99}, {5, 1511, 12042}, {98, 110, 132}, {399, 12188, 21667}, {441, 37987, 47085}, {3265, 47194, 47216}, {10620, 13188, 22337}, {12041, 33813, 49117}, {15054, 23235, 34549}, {36170, 56370, 65772}
X(72671) = pole of the line {161, 542} with respect to circumcircle
X(72671) = pole of the line {34842, 65770} with respect to Droz-Farny 1st circle
X(72671) = pole of the line {542, 6391} with respect to Droz-Farny 2nd circle
X(72671) = pole of the line {1368, 34841} with respect to nine-point circle
X(72671) = pole of the line {14356, 66925} with respect to orthocentroidal circle
X(72671) = pole of the line {107, 3563} with respect to polar circle
X(72671) = pole of the line {542, 61680} with respect to Warren reflection circle
X(72671) = pole of the line {38356, 71205} with respect to Brocard inellipse
X(72671) = pole of the line {6388, 6587} with respect to Kiepert hyperbola
X(72671) = pole of the line {20580, 32605} with respect to Kiepert parabola
X(72671) = pole of the line {22146, 23115} with respect to MacBeath circumconic
X(72671) = pole of the line {1368, 2972} with respect to MacBeath inconic
X(72671) = pole of the line {1562, 44518} with respect to orthic inconic
X(72671) = pole of the line {250, 7468} with respect to Stammler hyperbola
X(72671) = pole of the line {2693, 3563} with respect to Stammler reflection hyperbola
X(72671) = pole of the line {6527, 20080} with respect to Steiner circumellipse
X(72671) = pole of tripolar of X(17932) with respect to Steiner inellipse
X(72671) = pole of the line {107, 10425} with respect to Wallace hyperbola
X(72671) = pole of the line {542, 9919} with respect to orthoptic circle of circumcircle
X(72671) = pole of the line {35520, 65518} with respect to orthoptic circle of circumcircle of the Johnson triangle
X(72671) = pole of tripolar of X(15459) with respect to orthoptic circle of Jerabek hyperbola
X(72671) = pole of tripolar of X(65354) with respect to dual conic of polar circle
X(72671) = pole of the line {264, 62645} with respect to dual conic of Stammler hyperbola
X(72671) = pole of the line {4, 3566} with respect to dual conic of Wallace hyperbola
X(72671) = pole of tripolar of X(62645) with respect to dual conic of Moses HK-parabola
X(72671) = tripolar centroid of X(56267)
X(72671) = cross-difference of every pair of points on the line X(24)X(112)
X(72671) = foot of the perpendicular from X(i) to the line X(j)X(k) for these {i,j,k}: {44921, 4885, 5159}
X(72671) = barycentric product X(i)*X(j) for these (i, j): {114, 53173}, {230, 3265}, {339, 56389}, {460, 4143}, {520, 51481}, {525, 3564}, {879, 62590}, {1692, 52617}, {1733, 24018}, {2799, 53783}, {3267, 52144}, {3926, 55122}, {4176, 71409}, {4226, 15526}, {6333, 65726}, {6394, 55267}, {17216, 70566}, {17932, 41181}, {35067, 62645}, {36793, 61213}, {36875, 41077}, {39201, 70255}, {39473, 56687}, {42663, 70249}, {44145, 52613}, {52350, 57154}, {57109, 70564}, {68130, 70565}
X(72671) = barycentric quotient X(i)/X(j) for these (i, j): {3, 32697}, {63, 36105}, {69, 65354}, {125, 60338}, {230, 107}, {394, 10425}, {460, 6529}, {520, 2987}, {525, 35142}, {647, 3563}, {684, 57493}, {822, 36051}, {1648, 52476}, {1650, 65758}, {1692, 32713}, {1733, 823}, {2351, 58961}, {3265, 8781}, {3269, 35364}, {3564, 648}, {3926, 65277}, {4143, 57872}, {4226, 23582}, {5489, 66162}, {6394, 55266}, {8772, 24019}, {14417, 70199}, {15526, 62645}, {24018, 8773}, {24284, 47736}, {32320, 42065}, {35067, 4226}, {36875, 15459}, {39201, 32654}, {39473, 56572}, {41077, 36891}, {41181, 16230}, {42663, 2207}, {44145, 15352}, {47406, 4230}, {51335, 58070}, {51481, 6528}, {51610, 57071}, {51820, 20031}, {52144, 112}, {52613, 43705}, {53173, 40428}, {53783, 2966}, {55122, 393}, {55267, 6530}, {56389, 250}, {56687, 65265}, {57154, 11547}, {61213, 23964}, {62590, 877}, {62645, 57553}, {65726, 685}, {70565, 52919}, {71409, 6524}
X(72671) = trilinear product X(i)*X(j) for these (i, j): {230, 24018}, {326, 55122}, {520, 1733}, {656, 3564}, {822, 51481}, {1102, 71409}, {2632, 4226}, {3265, 8772}, {14208, 52144}, {17462, 53173}, {17879, 61213}, {20902, 56389}, {57109, 70565}, {70368, 70566}
X(72671) = trilinear quotient X(i)/X(j) for these (i, j): {63, 32697}, {69, 36105}, {107, 1733}, {162, 3564}, {230, 24019}, {304, 65354}, {326, 10425}, {520, 36051}, {656, 3563}, {822, 32654}, {823, 51481}, {1096, 55122}, {1820, 58961}, {2632, 35364}, {2987, 24018}, {3265, 8773}, {4226, 24000}, {8772, 32713}, {14208, 35142}, {17462, 58070}, {17879, 62645}, {20902, 60338}, {32676, 52144}, {36084, 53783}, {36092, 56687}, {36104, 65726}, {36126, 44145}, {52919, 70564}, {52920, 70565}, {57973, 70255}, {62590, 62720}
X(72671) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6334, 690}, {684, 65723, 68791}, {684, 68791, 9033}, {3265, 47194, 520}, {14417, 65723, 9033}, {14417, 68791, 684}

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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.