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(72682) = X(4)X(9864)∩X(98)X(740)

Antreas Hatzpolakis and Ercole Suppa, euclid 9788.

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}

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.