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(72692) = X(3)X(161)∩X(389)X(8883)

Barycentrics    a^2*(a^14-5*a^12*b^2+11*a^10*b^4-15*a^8*b^6+15*a^6*b^8-11*a^4*b^10+5*a^2*b^12-b^14-5*a^12*c^2+17*a^10*b^2*c^2-21*a^8*b^4*c^2+7*a^6*b^6*c^2+10*a^4*b^8*c^2-12*a^2*b^10*c^2+4*b^12*c^2+11*a^10*c^4-21*a^8*b^2*c^4+10*a^6*b^4*c^4+a^4*b^6*c^4+5*a^2*b^8*c^4-6*b^10*c^4-15*a^8*c^6+7*a^6*b^2*c^6+a^4*b^4*c^6+4*a^2*b^6*c^6+3*b^8*c^6+15*a^6*c^8+10*a^4*b^2*c^8+5*a^2*b^4*c^8+3*b^6*c^8-11*a^4*c^10-12*a^2*b^2*c^10-6*b^4*c^10+5*a^2*c^12+4*b^2*c^12-c^14) : :
X(72692) = X[3]+X[56308]

Antreas Hatzpolakis and Ercole Suppa, euclid 9804.

X(72692) lies on these lines: {3, 161}, {24, 53386}, {54, 52540}, {186, 19192}, {389, 8883}, {418, 18416}, {549, 23320}, {3432, 22268}, {6641, 61747}, {10274, 46025}, {10282, 23719}, {11202, 37813}, {14652, 66737}, {15869, 61753}, {18475, 34218}, {21394, 43809}, {32391, 58468}

X(72692) = midpoint of X(3) and X(56308)
X(72692) = pole of the line {3520, 6750} with respect to Stammler reflection hyperbola
X(72692) = {X(3),X(56308)}-harmonic conjugate of X(18400)

---

X(72693) = X(1)X(528)∩X(7)X(1155)

Barycentrics    a^5+2*a^4*b-9*a^3*b^2+5*a^2*b^3+4*a*b^4-3*b^5+2*a^4*c+4*a^3*b*c-5*a^2*b^2*c-10*a*b^3*c+9*b^4*c-9*a^3*c^2-5*a^2*b*c^2+12*a*b^2*c^2-6*b^3*c^2+5*a^2*c^3-10*a*b*c^3-6*b^2*c^3+4*a*c^4+9*b*c^4-3*c^5 : :
X(72693) = X[7]+X[5218], 4*X[60980]+X[61153]

Antreas Hatzpolakis and Ercole Suppa, euclid 9804.

X(72693) lies on these lines: {1, 528}, {2, 44785}, {3, 30424}, {7, 1155}, {65, 30340}, {142, 15346}, {516, 37606}, {1001, 17010}, {1156, 61008}, {1159, 5542}, {2646, 30332}, {3485, 3522}, {3812, 4208}, {4002, 13750}, {4795, 24980}, {4860, 30379}, {5087, 34919}, {5219, 5851}, {5220, 21075}, {5572, 25722}, {5729, 60978}, {6857, 15254}, {13464, 52682}, {15726, 17603}, {21620, 36279}, {30295, 34879}, {30331, 61279}, {35445, 38454}, {37298, 60905}, {37374, 55922}, {37541, 60993}, {59476, 60933}, {60946, 61648}, {60980, 61153}, {61035, 67097}

X(72693) = midpoint of X(7) and X(5218)
X(72693) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(3254),X(63166)}, {A,B,C,X(6173),X(37757)}, {A,B,C,X(42064),X(55920)}}
X(72693) = pole of the line {7671, 18839} with respect to Feuerbach hyperbola
X(72693) = pole of the line {1638, 30181} with respect to Steiner inellipse
X(72693) = pole of the line {527, 34522} with respect to dual conic of Yff parabola
X(72693) = cross-difference of every pair of points on the line X(17425)X(22108)
X(72693) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6173, 10427, 5880}, {43180, 64113, 36279}

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X(72694) = X(3)X(3626)∩X(9)X(80)

Barycentrics    a^4-5*a^3*b+2*a^2*b^2+5*a*b^3-3*b^4-5*a^3*c+14*a^2*b*c-11*a*b^2*c+2*a^2*c^2-11*a*b*c^2+6*b^2*c^2+5*a*c^3-3*c^4 : :
X(72694) = X[8]+X[59572], 5*X[3617]-X[5274]

Antreas Hatzpolakis and Ercole Suppa, euclid 9804.

X(72694) lies on these lines: {2, 44784}, {3, 3626}, {8, 1319}, {9, 80}, {10, 10912}, {40, 33559}, {57, 8256}, {355, 35460}, {484, 64087}, {518, 18419}, {1158, 5690}, {1532, 63143}, {2099, 6735}, {3057, 3617}, {3632, 13747}, {3748, 5554}, {3877, 58663}, {5123, 34640}, {5177, 5836}, {5880, 51782}, {6929, 38176}, {7504, 70802}, {10915, 15934}, {11362, 37822}, {11545, 68616}, {11715, 26446}, {12607, 18421}, {13462, 38455}, {13528, 59388}, {24703, 51433}, {25681, 63210}, {31508, 32157}, {34122, 64203}, {34647, 51362}, {43174, 52683}, {49169, 51788}, {63990, 66240}, {64204, 66257}

X(72694) = midpoint of X(8) and X(59572) for these {i,j}: {8, 59572}
X(72694) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(80),X(63167)}, {A,B,C,X(2161),X(63163)}}
X(72694) = cross-difference of every pair of points on the line X(17424)X(53314)
X(72694) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 37829, 37828}, {8, 59572, 33956}, {3679, 63137, 3036}

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X(72695) = X(7)X(11)∩X(10)X(141)

Barycentrics    a^3*b^2-3*a^2*b^3+3*a*b^4-b^5+4*a^3*b*c+3*a^2*b^2*c-10*a*b^3*c+3*b^4*c+a^3*c^2+3*a^2*b*c^2+14*a*b^2*c^2-2*b^3*c^2-3*a^2*c^3-10*a*b*c^3-2*b^2*c^3+3*a*c^4+3*b*c^4-c^5 : :
X(72695) = 3*X[4321]+X[5727], X[5289]-3*X[38053], 5*X[62778]+3*X[64151], X[11019]+X[61022], 5*X[31249]-X[36973]

Antreas Hatzpolakis and Ercole Suppa, euclid 9804.

X(72695) lies on the circumconic {{A,B,C,X(55076),X(62723)}} and these lines: {5, 43180}, {7, 11}, {10, 141}, {12, 30340}, {56, 390}, {57, 38454}, {354, 8255}, {496, 30424}, {527, 3816}, {528, 999}, {1001, 50739}, {1385, 43151}, {1418, 53617}, {2800, 20330}, {2886, 6173}, {3058, 30295}, {3333, 3813}, {3338, 4312}, {3660, 5572}, {3822, 51098}, {3873, 61035}, {3925, 59374}, {4321, 5727}, {4999, 38061}, {5045, 64113}, {5126, 30331}, {5218, 8732}, {5221, 60926}, {5289, 38053}, {5434, 45043}, {5559, 10390}, {5708, 60895}, {5854, 11041}, {5902, 38055}, {6067, 33108}, {6600, 35023}, {6601, 15179}, {7676, 41341}, {7951, 59372}, {8545, 17728}, {9710, 11037}, {10058, 42884}, {10177, 17051}, {10569, 61028}, {10589, 60967}, {11019, 15726}, {11036, 45085}, {11680, 59375}, {15008, 58573}, {15254, 64124}, {15587, 41573}, {15733, 58577}, {15888, 30312}, {17366, 67146}, {20116, 31657}, {21346, 68914}, {21617, 61660}, {21955, 63576}, {31249, 36973}, {34753, 60912}, {59476, 61019}, {60919, 60948}, {60980, 64443}, {60984, 72013}, {60988, 63258}, {60993, 64127}, {61015, 61649}, {63980, 65452}

X(72695) = midpoint of X(11019) and X(61022)
X(72695) = perspector of the circumconic through X(54118) and X(60487)
X(72695) = pole of the line {1638, 4905} with respect to incircle
X(72695) = pole of the line {3058, 15726} with respect to Feuerbach hyperbola
X(72695) = pole of the line {37, 1323} with respect to dual conic of Yff parabola
X(72695) = pole of the line {2886, 30379} with respect to dual conic of Moses-Feuerbach circumconic
X(72695) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {142, 15841, 58563}, {354, 30379, 8255}, {6173, 41555, 2886}, {11019, 61022, 15726}

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X(72696) = X(5)X(4691)∩X(10)X(141)

Barycentrics    a^2*b^2-b^4+4*a^2*b*c-10*a*b^2*c+a^2*c^2-10*a*b*c^2+2*b^2*c^2-c^4 : :
X(72696) = 7*X[9709]-3*X[19706], 3*X[3679]+X[15829]

Antreas Hatzpolakis and Ercole Suppa, euclid 9804.

X(72696) lies on the circumconic {{A,B,C,X(17758),X(42339)}} and these lines: {5, 4691}, {8, 1997}, {9, 32157}, {10, 141}, {11, 4678}, {12, 53620}, {145, 50038}, {200, 66257}, {210, 8256}, {496, 4669}, {528, 2551}, {529, 9709}, {936, 38455}, {958, 59591}, {1210, 4711}, {1329, 3679}, {1376, 37267}, {1706, 17768}, {2885, 3840}, {2886, 3614}, {3436, 49732}, {3452, 13463}, {3625, 17527}, {3626, 3813}, {3633, 17575}, {3711, 5554}, {3740, 6736}, {3921, 10039}, {3968, 6147}, {3983, 6735}, {4015, 5690}, {4187, 4668}, {4413, 56879}, {4679, 63142}, {4745, 31419}, {5084, 8168}, {5253, 34689}, {5316, 66256}, {5815, 5852}, {5837, 58629}, {6667, 64081}, {6690, 7080}, {7681, 59503}, {8165, 11235}, {8580, 32049}, {9708, 64123}, {9710, 17757}, {10107, 21060}, {11260, 20103}, {12447, 32537}, {15481, 43174}, {15888, 46933}, {17051, 25011}, {18236, 27870}, {21896, 66071}, {25639, 38098}, {26105, 66259}, {26127, 34699}, {30393, 64204}, {32198, 46694}, {34606, 36005}, {40257, 61628}, {44720, 69091}, {44847, 51072}, {50244, 57288}, {51415, 59310}, {61510, 63980}, {62061, 63751}

X(72696) = pole of the line {2886, 6736} with respect to dual conic of Moses-Feuerbach circumconic
X(72696) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 9711, 3816}, {10, 12607, 3826}, {3617, 21031, 2886}, {3626, 3820, 3813}

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X(72697) = X(3)X(66)∩X(20)X(5893)

Barycentrics    10*a^10-21*a^8*b^2+4*a^6*b^4+14*a^4*b^6-6*a^2*b^8-b^10-21*a^8*c^2+32*a^6*b^2*c^2-14*a^4*b^4*c^2+3*b^8*c^2+4*a^6*c^4-14*a^4*b^2*c^4+12*a^2*b^4*c^4-2*b^6*c^4+14*a^4*c^6-2*b^4*c^6-6*a^2*c^8+3*b^2*c^8-c^10 : :
X(72697) = 5*X[3]-X[6247], 3*X[3]-X[6696], 7*X[3]+X[9833], 9*X[3]-X[14216], 7*X[3]-3*X[23328], 3*X[3]+X[34782], 3*X[6247]-5*X[6696], 7*X[6247]+5*X[9833], 9*X[6247]-5*X[14216], 7*X[6247]-15*X[23328], 3*X[6247]+5*X[34782], 3*X[6247]+X[64033], 7*X[6696]+3*X[9833], 3*X[6696]-X[14216], 7*X[6696]-9*X[23328], X[6696]+X[34782], 5*X[6696]+X[64033], 9*X[9833]+7*X[14216], X[9833]+3*X[23328], 3*X[9833]-7*X[34782], 7*X[14216]-27*X[23328], X[14216]+3*X[34782], 5*X[14216]+3*X[64033], X[15585]+X[44882], 3*X[21167]+X[36989], 9*X[23328]+7*X[34782], 5*X[34782]-X[64033], 3*X[61683]+X[64196], X[4]-3*X[58434], X[20]+X[5893], X[20]+3*X[10192], 5*X[20]+3*X[61721], 3*X[20]+5*X[64024], 3*X[20]+X[68058], X[5893]-3*X[10192], 5*X[5893]-3*X[61721], 3*X[5893]-5*X[64024], 3*X[5893]-X[68058], 5*X[10192]-X[61721], 9*X[10192]-5*X[64024], 9*X[10192]-X[68058], 9*X[61721]-25*X[64024], 9*X[61721]-5*X[68058], 5*X[64024]-X[68058], X[32903]+X[64063], X[64]-9*X[10304], 3*X[140]-X[18383], 3*X[154]+5*X[3522], 3*X[154]+X[5894], 9*X[154]-X[6225]

Antreas Hatzpolakis and Ercole Suppa, euclid 9804.

X(72697) lies on these lines: {3, 66}, {4, 58434}, {20, 5893}, {30, 32903}, {64, 10304}, {140, 18383}, {154, 3522}, {184, 68907}, {186, 12241}, {206, 46374}, {343, 38438}, {376, 2883}, {378, 16656}, {389, 37934}, {546, 10182}, {548, 10282}, {549, 34785}, {550, 11202}, {631, 41362}, {632, 34786}, {1192, 8550}, {1350, 53050}, {1498, 3528}, {1620, 6776}, {1657, 67868}, {1853, 15717}, {1885, 15448}, {2393, 17704}, {2777, 13392}, {2781, 13348}, {2888, 38448}, {3146, 61680}, {3183, 15576}, {3357, 46853}, {3520, 16621}, {3523, 17845}, {3524, 64037}, {3525, 18405}, {3526, 23324}, {3530, 18400}, {3534, 51491}, {3819, 68028}, {5596, 55651}, {5656, 62092}, {5878, 62100}, {5895, 35260}, {5944, 47335}, {6000, 32142}, {6146, 21844}, {6240, 7699}, {6411, 8991}, {6412, 13980}, {6689, 31833}, {6759, 8703}, {6803, 18382}, {7499, 20376}, {8567, 11206}, {9729, 44668}, {9786, 12007}, {9820, 44242}, {10117, 16661}, {10170, 63728}, {10193, 61790}, {10298, 68018}, {10299, 40686}, {10541, 23326}, {10565, 27082}, {10606, 21735}, {10984, 44268}, {11204, 62069}, {11425, 37460}, {11430, 11745}, {12024, 26879}, {12100, 20299}, {12103, 61749}, {12108, 32767}, {12250, 62084}, {12289, 15153}, {12293, 66584}, {12315, 14093}, {12324, 62067}, {13093, 62075}, {13346, 64061}, {13367, 13568}, {13403, 37935}, {13567, 15750}, {14530, 62085}, {14862, 41981}, {14864, 61784}, {14869, 23325}, {15051, 23315}, {15105, 32063}, {15152, 26882}, {15246, 32345}, {15274, 34286}, {15331, 44665}, {15332, 61608}, {15583, 53094}, {15644, 41589}, {15646, 22962}

X(72697) = midpoint of X(i) and X(j) for these {i,j}: {20, 5893}, {548, 10282}, {550, 16252}, {5894, 68025}, {6696, 34782}, {9820, 44242}, {12103, 61749}, {15332, 61608}, {15585, 44882}, {15644, 41589}, {32903, 64063}
X(72697) = reflection of X(i) in X(j) for these {i,j}: {32184, 17704}, {32767, 12108}
X(72697) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(66),X(31361)}, {A,B,C,X(14376),X(37878)}}
X(72697) = center of circle {X(20), X(5893), X(67662)}
X(72697) = pole of the line {22, 8567} with respect to Stammler hyperbola
X(72697) = pole of the line {378, 1620} with respect to Stammler reflection hyperbola
X(72697) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 9833, 23328}, {3, 34782, 6696}, {20, 5893, 50709}, {20, 10192, 5893}, {20, 64024, 68058}, {154, 3522, 5894}, {154, 5894, 68025}, {376, 17821, 2883}, {548, 10282, 15311}, {550, 11202, 16252}, {3523, 17845, 23332}, {6247, 34782, 64033}, {6696, 34782, 1503}, {10192, 68058, 64024}, {11206, 21734, 8567}, {13367, 37931, 13568}, {15585, 44882, 1503}, {32903, 64063, 30}, {35260, 5X(72697) 0693, 5895}, {64024, 68058, 5893}

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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.