ΜΑΘΗΜΑΤΙΚΑ

Δευτέρα 1 Ιουνίου 2026

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(72681) = X(4)X(5851)∩X(80)X(527)

Barycentrics a*(a-b-c)*(a^3+2*a^2*b-7*a*b^2+4*b^3-a^2*c+6*a*b*c-7*b^2*c-a*c^2+2*b*c^2+c^3)*(a^3-a^2*b-a*b^2+b^3+2*a^2*c+6*a*b*c+2*b^2*c-7*a*c^2-7*b*c^2+4*c^3) : :
X(72681) = 2*X[11]-X[34919], X[100]-2*X[15346], X[20085]+3*X[60998], 5*X[31272]-4*X[63643]

Antreas Hatzpolakis and Ercole Suppa, euclid 9788.

X(72681) lies on the Feuerbach circumhyperbola and these lines: {4, 5851}, {7, 10707}, {8, 5856}, {9, 60782}, {11, 34919}, {80, 527}, {100, 15346}, {104, 15726}, {516, 64330}, {518, 24297}, {528, 1000}, {1156, 3218}, {1320, 15733}, {2320, 53055}, {2346, 17668}, {2801, 3577}, {3689, 34894}, {3738, 23893}, {9814, 11570}, {12755, 55924}, {13143, 61030}, {14497, 42871}, {15175, 64154}, {15909, 60962}, {18490, 25558}, {20085, 60998}, {30513, 45043}, {31272, 63643}, {34742, 55964}, {38307, 60884}, {38454, 64290}

X(72681) = reflection of X(i) in X(j) for these {i,j}: {100, 15346}, {34919, 11}
X(72681) = antigonal conjugate of X(34919)
X(72681) = symgonal image of X(15346)
X(72681) = trilinear pole of line {X(650), X(34522)}
X(72681) = lies on circumconics with center X(i) for i in {11, 15346}
X(72681) = lies on all circumconics with perspector on the line {650, 34522}
X(72681) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(1),X(4)}, {A,B,C,X(100),X(35157)}, {A,B,C,X(105),X(20119)}, {A,B,C,X(284),X(28535)}, {A,B,C,X(513),X(5856)}, {A,B,C,X(521),X(5851)}, {A,B,C,X(522),X(41798)}, {A,B,C,X(527),X(3218)}, {A,B,C,X(673),X(60782)}, {A,B,C,X(840),X(2316)}}
X(72681) = barycentric product X(4391)*X(53887)
X(72681) = barycentric quotient X(i)/X(j) for these (i,j): {{1, 50573}, {43065, 18801}, {53887, 651}
X(72681) = trilinear product X(522)*X(53887)
X(72681) = trilinear quotient X(i)/X(j) for these (i,j): {2, 50573}, {109, 53887}, {18801, 26015}

X(72682) = X(4)X(9864)∩X(98)X(740)

Barycentrics (b+c)*(a^4*b-a^2*b^3+a*b^4+b^5+b^4*c-a^3*c^2-a^2*b*c^2-b^3*c^2-a^2*c^3+b*c^4)*(-(a^3*b^2)-a^2*b^3+a^4*c-a^2*b^2*c+b^4*c-a^2*c^3-b^2*c^3+a*c^4+b*c^4+c^5) : :
X(72682) = X[99]-2*X[15349], 2*X[115]-X[43677]

Antreas Hatzpolakis and Ercole Suppa, euclid 9788.

X(72682) lies on the Kiepert circumhyperbola and these lines: {2, 71237}, {4, 9864}, {98, 740}, {99, 15349}, {115, 43677}, {516, 54546}, {519, 55003}, {537, 54605}, {542, 54609}, {752, 54492}, {2784, 3429}, {2796, 60172}, {2799, 4444}, {4697, 14534}, {20437, 40017}, {28580, 54491}, {32014, 59509}, {35103, 54563}, {54119, 66678}, {60320, 70729}

X(72682) = reflection of X(i) in X(j) for these {i,j}: {99, 15349}, {43677, 115}
X(72682) = antigonal conjugate of X(43677)
X(72682) = symgonal image of X(15349)
X(72682) = antitomic conjugate of X(43677)
X(72682) = trilinear pole of line {X(523), X(34528)}
X(72682) = lies on circumconics with center X(i) for i in {115, 15349}
X(72682) = lies on all circumconics with perspector on the line {523, 34528}
X(72682) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(2),X(4)}, {A,B,C,X(740),X(2799)}, {A,B,C,X(1934),X(7235)}, {A,B,C,X(3027),X(35544)}, {A,B,C,X(4647),X(4697)}}
X(72682) = barycentric quotient X(10026)/X(25607)
X(72682) = trilinear quotient X(25607)/X(68991)

X(72683) = X(40)X(29374)∩X(56)X(52315)

Barycentrics (b-c)^2*(-a+b+c)*(a^5-a^4*b-a*b^4+b^5-a^4*c+3*a^3*b*c-4*a^2*b^2*c+3*a*b^3*c-b^4*c-2*a^3*c^2+2*a^2*b*c^2+2*a*b^2*c^2-2*b^3*c^2+2*a^2*c^3-5*a*b*c^3+2*b^2*c^3+a*c^4+b*c^4-c^5)*(-a^5+a^4*b+2*a^3*b^2-2*a^2*b^3-a*b^4+b^5+a^4*c-3*a^3*b*c-2*a^2*b^2*c+5*a*b^3*c-b^4*c+4*a^2*b*c^2-2*a*b^2*c^2-2*b^3*c^2-3*a*b*c^3+2*b^2*c^3+a*c^4+b*c^4-c^5) : :
X(72683) = 2*X[3035]-X[43353]

Antreas Hatzpolakis and Ercole Suppa, euclid 9788.

X(72683) lies on the cubics K806, K826 and these lines: {40, 29374}, {56, 52315}, {3035, 43353}, {3436, 35313}, {11247, 35604}, {18340, 38560}, {21105, 35015}

X(72683) = reflection of X(43353) in X(3035)
X(72683) = isogonal conjugate of X(57105)
X(72683) = antigonal conjugate of X(11)
X(72683) = symgonal image of X(3035)
X(72683) = Miquel associate of X(68357)
X(72683) = trilinear pole of line {X(46101), X(52316)}
X(72683) = foot of the perpendicular from X(i) to the line X(j)X(k) for these {i,j,k}: {21105, 40, 29374}, {40, 21105, 35015}
X(72683) = lies on circumconic with center X(3035)
X(72683) = lies on all circumconics with perspector on the line {46101, 52316}
X(72683) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(1),X(1146)}, {A,B,C,X(4),X(11)}, {A,B,C,X(40),X(38357)}, {A,B,C,X(80),X(34896)}, {A,B,C,X(885),X(21105)}, {A,B,C,X(1086),X(64980)}, {A,B,C,X(1118),X(1358)}, {A,B,C,X(3160),X(68914)}, {A,B,C,X(4225),X(38345)}, {A,B,C,X(4534),X(56940)}}
X(72683) = barycentric product X(i)*X(j) for these (i,j): {11, 68357}, {4858, 29374}
X(72683) = barycentric quotient X(i)/X(j) for these (i,j): {6, 57105}, {11, 37781}, {2170, 1768}, {8735, 60356}, {29374, 4564}, {68357, 4998}
X(72683) = trilinear product X(i)*X(j) for these (i,j): {11, 29374}, {2170, 68357}
X(72683) = trilinear quotient X(i)/X(j) for these (i,j): {11, 1768}, {59, 29374}, {4564, 68357}, {4858, 37781}, {34345, 35015}

X(72684) = X(12)X(52119)∩X(442)X(5620)

Barycentrics (b+c)^2*(a^3+a^2*b+a*b^2+b^3-a^2*c-a*b*c-b^2*c-a*c^2-b*c^2+c^3)*(a^3-a^2*b-a*b^2+b^3+a^2*c-a*b*c-b^2*c+a*c^2-b*c^2+c^3)*(a^4-2*a^2*b^2+b^4-a^2*b*c-a*b^2*c-2*a^2*c^2-3*a*b*c^2-2*b^2*c^2+c^4)*(a^4-2*a^2*b^2+b^4-a^2*b*c-3*a*b^2*c-2*a^2*c^2-a*b*c^2-2*b^2*c^2+c^4) : :
X(72684) = X[12]-2*X[52119], 2*X[4999]-X[43354]

Antreas Hatzpolakis and Ercole Suppa, euclid 9788.

X(72684) lies on these lines: {12, 52119}, {442, 5620}, {4999, 43354}, {5535, 45926}, {5842, 51760}, {11604, 46441}

X(72684) = reflection of X(i) in X(j) for these {i,j}: {12, 52119}, {43354, 4999}
X(72684) = antigonal conjugate of X(12)
X(72684) = symgonal image of X(4999)
X(72684) = lies on circumconics with center X(i) for i in {4999, 52119}
X(72684) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(4),X(12)}, {A,B,C,X(65),X(46441)}, {A,B,C,X(502),X(42005)}, {A,B,C,X(4036),X(24298)}, {A,B,C,X(5535),X(68765)}, {A,B,C,X(43354),X(65281)}}
X(72684) = barycentric product X(12)*X(68358)
X(72684) = barycentric quotient X(68358)/X(261)
X(72684) = trilinear product X(2171)*X(68358)
X(72684) = trilinear quotient X(i)/X(j) for these (i,j): {2185, 68358}, {5620, 46441}

X(72685) = X(1)X(60987)∩X(2)X(15348)

Barycentrics (a-b-c)*(a^5+a^4*b-2*a^3*b^2-2*a^2*b^3+a*b^4+b^5-3*a^4*c-4*a^3*b*c+6*a^2*b^2*c-4*a*b^3*c-3*b^4*c+2*a^3*c^2+2*a^2*b*c^2+2*a*b^2*c^2+2*b^3*c^2+2*a^2*c^3+4*a*b*c^3+2*b^2*c^3-3*a*c^4-3*b*c^4+c^5)*(a^5-3*a^4*b+2*a^3*b^2+2*a^2*b^3-3*a*b^4+b^5+a^4*c-4*a^3*b*c+2*a^2*b^2*c+4*a*b^3*c-3*b^4*c-2*a^3*c^2+6*a^2*b*c^2+2*a*b^2*c^2+2*b^3*c^2-2*a^2*c^3-4*a*b*c^3+2*b^2*c^3+a*c^4-3*b*c^4+c^5) : :
X(72685) = 3*X[2]-2*X[15348]

Antreas Hatzpolakis and Ercole Suppa, euclid 9788.

X(72685) lies on the Feuerbach circumhyperbola and these lines: {1, 60987}, {2, 15348}, {4, 15733}, {21, 5766}, {72, 55964}, {84, 527}, {104, 5759}, {329, 1156}, {516, 56273}, {518, 3427}, {943, 63643}, {1260, 34894}, {2346, 60959}, {3062, 61010}, {3577, 5853}, {5809, 30513}, {5851, 34256}, {6172, 55960}, {7091, 60982}, {8058, 23893}, {10309, 15726}, {10429, 12528}, {12848, 56262}, {24389, 42015}, {38308, 63970}, {47387, 63168}, {55922, 61011}, {61030, 64265}

X(72685) = anticomplement of X(15348)
X(72685) = perspector of the inconic with center X(34526)
X(72685) = lies on circumconics with center X(i) for i in {11, 43960}
X(72685) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(1),X(4)}, {A,B,C,X(2),X(60987)}, {A,B,C,X(55),X(1242)}, {A,B,C,X(68),X(34902)}, {A,B,C,X(142),X(60959)}, {A,B,C,X(144),X(61010)}, {A,B,C,X(226),X(60997)}, {A,B,C,X(281),X(58002)}, {A,B,C,X(329),X(527)}, {A,B,C,X(346),X(39695)}}
X(72685) = barycentric product X(4391)*X(30237)
X(72685) = barycentric quotient X(i)/X(j) for these (i,j): {9, 1998}, {220, 47387}, {650, 30199}, {30237, 651}, {34526, 15348}
X(72685) = trilinear product X(522)*X(30237)
X(72685) = trilinear quotient X(i)/X(j) for these (i,j): {8, 1998}, {109, 30237}, {200, 47387}, {522, 30199}

X(72686) = X(28)X(3556)∩X(34)X(4331)

Barycentrics (a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^5+a^4*b-2*a^3*b^2-2*a^2*b^3+a*b^4+b^5-a^4*c-2*a^3*b*c+2*a^2*b^2*c-2*a*b^3*c-b^4*c+2*a*b*c^3-a*c^4-b*c^4+c^5)*(a^5-a^4*b-a*b^4+b^5+a^4*c-2*a^3*b*c+2*a*b^3*c-b^4*c-2*a^3*c^2+2*a^2*b*c^2-2*a^2*c^3-2*a*b*c^3+a*c^4-b*c^4+c^5) : :

Antreas Hatzpolakis and Ercole Suppa, euclid 9788.

X(72686) lies on these lines: {4, 44670}, {28, 3556}, {34, 4331}, {286, 4329}, {915, 44059}, {946, 55105}, {1119, 64747}, {5317, 21148}

X(72686) = lies on circumconics with center X(i) for i in {5521, 43963}
X(72686) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(4),X(19)}, {A,B,C,X(7),X(393)}, {A,B,C,X(25),X(513)}, {A,B,C,X(66),X(55024)}, {A,B,C,X(1002),X(8749)}, {A,B,C,X(1042),X(3556)}, {A,B,C,X(1093),X(58007)}, {A,B,C,X(1857),X(14775)}, {A,B,C,X(1880),X(15320)}, {A,B,C,X(2217),X(43695)}}
X(72686) = barycentric product X(17924)*X(44059)
X(72686) = barycentric quotient X(i)/X(j) for these (i,j): {19, 2000}, {34, 2002}, {6591, 8760}, {44059, 1332}
X(72686) = trilinear product X(7649)*X(44059)
X(72686) = trilinear quotient X(i)/X(j) for these (i,j): {4, 2000}, {278, 2002}, {1331, 44059}, {7649, 8760}

X(72687) = X(2)X(35888)∩X(30)X(15345)

Barycentrics (a^4-2*a^2*b^2+b^4-2*a^2*c^2-b^2*c^2+c^4)*(2*a^12-5*a^10*b^2+2*a^8*b^4+5*a^6*b^6-7*a^4*b^8+4*a^2*b^10-b^12-5*a^10*c^2+2*a^8*b^2*c^2+4*a^6*b^4*c^2+5*a^4*b^6*c^2-12*a^2*b^8*c^2+6*b^10*c^2+2*a^8*c^4+4*a^6*b^2*c^4+4*a^4*b^4*c^4+8*a^2*b^6*c^4-15*b^8*c^4+5*a^6*c^6+5*a^4*b^2*c^6+8*a^2*b^4*c^6+20*b^6*c^6-7*a^4*c^8-12*a^2*b^2*c^8-15*b^4*c^8+4*a^2*c^10+6*b^2*c^10-c^12) : :
X(72687) = 3*X[2]-X[35888], X[5]-2*X[23338], X[546]-2*X[31879], X[140]+X[23337], 3*X[547]-4*X[15425], 2*X[3628]-X[14143], 5*X[12812]-6*X[34597], X[28237]+X[35720], X[32551]-2*X[34598]

Antreas Hatzpolakis and Ercole Suppa, euclid 9788.

X(72687) lies on these lines: {2, 35888}, {5, 23338}, {30, 15345}, {54, 24573}, {110, 13362}, {113, 137}, {140, 6150}, {428, 35887}, {547, 15425}, {1141, 15307}, {1154, 33545}, {3628, 14143}, {5501, 50708}, {7745, 39018}, {8902, 32165}, {10024, 63837}, {11819, 62589}, {12605, 34833}, {12812, 34597}, {13163, 32639}, {20414, 25150}, {28237, 35720}, {32551, 34598}

X(72687) = midpoint of X(i) and X(j) for these {i,j}: {140, 23337}, {28237, 35720}
X(72687) = reflection of X(i) in X(j) for these {i,j}: {5, 23338}, {546, 31879}, {14143, 3628}, {32551, 34598}
X(72687) = complement of X(35888)
X(72687) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {140, 23337, 32744}, {6150, 31376, 140}

X(72688) = X(2)X(8547)∩X(5)X(2854)

Barycentrics 3*a^6*b^2-a^4*b^4+3*a^2*b^6+b^8-3*a^6*c^2+2*a^4*b^2*c^2-9*a^2*b^4*c^2-a^4*c^4-9*a^2*b^2*c^4-2*b^4*c^4+3*a^2*c^6+c^8 : :
X(72688) = 3*X[2]-X[8547], 3*X[5]-2*X[20113], X[6]-2*X[25488], 2*X[3589]-X[8546], X[3818]+X[8542], X[5092]-2*X[63648], 2*X[32154]-X[48906], 2*X[37283]-X[44882], 3*X[50977]-X[66615]

Antreas Hatzpolakis and Ercole Suppa, euclid 9788.

X(72688) lies on these lines: {2, 8547}, {5, 2854}, {6, 25488}, {110, 597}, {125, 29959}, {141, 858}, {381, 524}, {542, 12039}, {599, 31105}, {1503, 50008}, {2393, 20300}, {2930, 14389}, {3589, 8546}, {3631, 3867}, {3818, 8542}, {4549, 29181}, {5092, 63648}, {5169, 62381}, {5648, 53843}, {7495, 12367}, {7533, 41617}, {8550, 15037}, {9019, 41714}, {9027, 19130}, {9145, 37345}, {10546, 47455}, {14094, 14982}, {15018, 32255}, {15435, 18919}, {15583, 34573}, {16042, 25320}, {18553, 25489}, {20582, 32216}, {22165, 41721}, {22330, 45184}, {23410, 44490}, {25328, 37648}, {32111, 47353}, {32154, 48906}, {33851, 44218}, {35001, 48881}, {36990, 44440}, {37283, 44882}, {50977, 66615}, {50991, 71230}

X(72688) = midpoint of X(3818) and X(8542)
X(72688) = reflection of X(i) in X(j) for these {i,j}: {6, 25488}, {5092, 63648}, {8546, 3589}, {44882, 37283}, {48906, 32154}
X(72688) = complement of X(8547)
X(72688) = center of circle {X(3734), X(3818), X(8542)}
X(72688) = pole of the line {2549, 9465} with respect to Kiepert hyperbola
X(72688) = pole of the line {26233, 69450} with respect to Wallace hyperbola
X(72688) = pole of the line {2780, 68787} with respect to orthoptic circle of nine-point circle

X(72689) = X(11)X(8261)∩X(442)X(10572)

Barycentrics a^11*b^2-3*a^10*b^3-a^9*b^4+11*a^8*b^5-6*a^7*b^6-14*a^6*b^7+14*a^5*b^8+6*a^4*b^9-11*a^3*b^10+a^2*b^11+3*a*b^12-b^13-2*a^11*b*c+3*a^10*b^2*c+5*a^9*b^3*c-11*a^8*b^4*c+2*a^7*b^5*c+12*a^6*b^6*c-16*a^5*b^7*c+16*a^3*b^9*c-7*a^2*b^10*c-5*a*b^11*c+3*b^12*c+a^11*c^2+3*a^10*b*c^2-8*a^9*b^2*c^2+2*a^8*b^3*c^2+12*a^7*b^4*c^2-12*a^6*b^5*c^2-12*a^5*b^6*c^2+8*a^4*b^7*c^2+15*a^3*b^8*c^2-3*a^2*b^9*c^2-8*a*b^10*c^2+2*b^11*c^2-3*a^10*c^3+5*a^9*b*c^3+2*a^8*b^2*c^3-20*a^7*b^3*c^3+8*a^6*b^4*c^3+16*a^5*b^5*c^3-14*a^4*b^6*c^3-16*a^3*b^7*c^3+21*a^2*b^8*c^3+15*a*b^9*c^3-14*b^10*c^3-a^9*c^4-11*a^8*b*c^4+12*a^7*b^2*c^4+8*a^6*b^3*c^4-16*a^5*b^4*c^4-4*a^4*b^5*c^4-4*a^3*b^6*c^4+10*a^2*b^7*c^4+5*a*b^8*c^4+5*b^9*c^4+11*a^8*c^5+2*a^7*b*c^5-12*a^6*b^2*c^5+16*a^5*b^3*c^5-4*a^4*b^4*c^5-22*a^2*b^6*c^5-10*a*b^7*c^5+25*b^8*c^5-6*a^7*c^6+12*a^6*b*c^6-12*a^5*b^2*c^6-14*a^4*b^3*c^6-4*a^3*b^4*c^6-22*a^2*b^5*c^6-20*b^7*c^6-14*a^6*c^7-16*a^5*b*c^7+8*a^4*b^2*c^7-16*a^3*b^3*c^7+10*a^2*b^4*c^7-10*a*b^5*c^7-20*b^6*c^7+14*a^5*c^8+15*a^3*b^2*c^8+21*a^2*b^3*c^8+5*a*b^4*c^8+25*b^5*c^8+6*a^4*c^9+16*a^3*b*c^9-3*a^2*b^2*c^9+15*a*b^3*c^9+5*b^4*c^9-11*a^3*c^10-7*a^2*b*c^10-8*a*b^2*c^10-14*b^3*c^10+a^2*c^11-5*a*b*c^11+2*b^2*c^11+3*a*c^12+3*b*c^12-c^13 : :

Antreas Hatzpolakis and Ercole Suppa, euclid 9788.

X(72689) lies on these lines: {11, 8261}, {442, 10572}, {758, 63963}, {6882, 54212}, {44669, 64274}

X(72690) = X(5)X(675)∩X(118)X(381)

Barycentrics a^10-a^9*b-a^8*b^2+2*a^7*b^3-3*a^6*b^4+2*a^5*b^5+2*a^4*b^6-4*a^3*b^7+2*a^2*b^8+a*b^9-b^10-a^9*c+a^8*b*c+2*a^5*b^4*c-2*a^4*b^5*c-a*b^8*c+b^9*c-a^8*c^2-4*a^6*b^2*c^2+4*a^5*b^3*c^2-3*a^4*b^4*c^2+4*a^3*b^5*c^2-a^2*b^6*c^2-2*a*b^7*c^2+3*b^8*c^2+2*a^7*c^3+4*a^5*b^2*c^3-2*a^4*b^3*c^3-2*a^2*b^5*c^3+2*a*b^6*c^3-4*b^7*c^3-3*a^6*c^4+2*a^5*b*c^4-3*a^4*b^2*c^4+2*a^2*b^4*c^4-2*b^6*c^4+2*a^5*c^5-2*a^4*b*c^5+4*a^3*b^2*c^5-2*a^2*b^3*c^5+6*b^5*c^5+2*a^4*c^6-a^2*b^2*c^6+2*a*b^3*c^6-2*b^4*c^6-4*a^3*c^7-2*a*b^2*c^7-4*b^3*c^7+2*a^2*c^8-a*b*c^8+3*b^2*c^8+a*c^9+b*c^9-c^10 : :
X(72690) = X[3]-2*X[5513], 2*X[5]-X[675], X[44876]+X[44968], 3*X[381]-2*X[25642], 5*X[1656]-4*X[31380]

Antreas Hatzpolakis and Ercole Suppa, euclid 9788.

X(72690) lies on the circumcircle of the Johnson triangle and these lines: {3, 5513}, {5, 675}, {30, 44876}, {118, 381}, {1478, 33966}, {1479, 6025}, {1656, 31380}, {5779, 10747}, {10738, 20430}, {10749, 68365}, {18531, 34335}

X(72690) = midpoint of X(44876) and X(44968)
X(72690) = reflection of X(i) in X(j) for these {i,j}: {3, 5513}, {675, 5}
X(72690) = reflection of X(53190) in the line X(5)X(523)

X(72691) = X(3)X(35971)∩X(5)X(689)

Barycentrics a^2*(a^6*b^8-a^4*b^10+a^10*b^2*c^2-a^8*b^4*c^2-a^6*b^6*c^2+4*a^4*b^8*c^2+a^2*b^10*c^2-b^12*c^2-a^8*b^2*c^4-2*a^6*b^4*c^4-a^4*b^6*c^4+3*b^10*c^4-a^6*b^2*c^6-a^4*b^4*c^6-a^2*b^6*c^6-3*b^8*c^6+a^6*c^8+4*a^4*b^2*c^8-3*b^6*c^8-a^4*c^10+a^2*b^2*c^10+3*b^4*c^10-b^2*c^12) : :
X(72691) = X[3]-2*X[35971], 2*X[5]-X[689], X[44937]+X[53889], 3*X[381]-2*X[44947]

Antreas Hatzpolakis and Ercole Suppa, euclid 9788.

X(72691) lies on the circumcircle of the Johnson triangle and these lines: {3, 35971}, {5, 689}, {30, 37888}, {381, 44947}, {1478, 7334}, {1479, 6026}, {9301, 66827}

X(72691) = midpoint of X(44937) and X(53889) for these {i,j}: {44937, 53889}
X(72691) = reflection of X(i) in X(j) for these {i,j}: {3, 35971}, {689, 5}
X(72691) = reflection of X(53918) in the line X(5)X(523)

Παρασκευή 22 Μαΐου 2026

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

Πέμπτη 21 Μαΐου 2026

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.