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(72487) = X(3)X(476)∩X(30)X(5972)

Barycentrics   4*a^16-11*a^14*b^2-a^12*b^4+30*a^10*b^6-30*a^8*b^8+a^6*b^10+11*a^4*b^12-4*a^2*b^14-11*a^14*c^2+50*a^12*b^2*c^2-56*a^10*b^4*c^2-22*a^8*b^6*c^2+68*a^6*b^8*c^2-31*a^4*b^10*c^2+3*a^2*b^12*c^2-b^14*c^2-a^12*c^4-56*a^10*b^2*c^4+132*a^8*b^4*c^4-72*a^6*b^6*c^4-24*a^4*b^8*c^4+15*a^2*b^10*c^4+6*b^12*c^4+30*a^10*c^6-22*a^8*b^2*c^6-72*a^6*b^4*c^6+88*a^4*b^6*c^6-14*a^2*b^8*c^6-15*b^10*c^6-30*a^8*c^8+68*a^6*b^2*c^8-24*a^4*b^4*c^8-14*a^2*b^6*c^8+20*b^8*c^8+a^6*c^10-31*a^4*b^2*c^10+15*a^2*b^4*c^10-15*b^6*c^10+11*a^4*c^12+3*a^2*b^2*c^12+6*b^4*c^12-4*a^2*c^14-b^2*c^14 : :
X(72487) = 3*X[2]-X[66778], 5*X[3]-X[476], 3*X[3]+X[477], 9*X[3]-X[38580], 7*X[3]+X[38581], 3*X[3]-X[38609], X[3]+X[38610], 7*X[3]-3*X[38700], X[3]+3*X[38701], 2*X[3]-X[68307], 3*X[476]+5*X[477], 9*X[476]-5*X[38580], 7*X[476]+5*X[38581], 3*X[476]-5*X[38609], X[476]+5*X[38610], 7*X[476]-15*X[38700], 2*X[476]-5*X[68307], 3*X[477]+X[38580], 7*X[477]-3*X[38581], X[477]+X[38609], X[477]-3*X[38610], 7*X[477]+9*X[38700], X[477]-9*X[38701], 2*X[477]+3*X[68307], 7*X[38580]+9*X[38581], X[38580]-3*X[38609], X[38580]+9*X[38610], 7*X[38580]-27*X[38700], 2*X[38580]-9*X[68307], 3*X[38581]+7*X[38609], X[38581]-7*X[38610], X[38581]+3*X[38700], X[38581]-21*X[38701], 2*X[38581]+7*X[68307], X[38609]+3*X[38610], 7*X[38609]-9*X[38700], X[38609]+9*X[38701], 2*X[38609]-3*X[68307], 7*X[38610]+3*X[38700], X[38610]-3*X[38701], 2*X[38610]+X[68307], 7*X[38677]-39*X[38700], 2*X[38677]-13*X[68307], X[38678]-33*X[38701], X[38700]+7*X[38701], 6*X[38700]-7*X[68307], 6*X[38701]+X[68307], X[20]+3*X[57306], X[20]+X[66795], 3*X[57306]-X[66795]

Antreas Hatzipolakis and Ercole Suppa, euclid 9633.

X(72487) lies on these lines: {2, 66778}, {3, 476}, {20, 57306}, {30, 5972}, {140, 64510}, {186, 66771}, {376, 20957}, {382, 66801}, {541, 33505}, {549, 25641}, {550, 3258}, {631, 66781}, {1511, 36164}, {1656, 14989}, {3520, 66790}, {3523, 57305}, {3524, 34193}, {3528, 14731}, {3529, 66819}, {3530, 22104}, {3534, 44967}, {3579, 66770}, {3628, 68308}, {5010, 33964}, {5054, 66772}, {5122, 59825}, {5663, 47084}, {6723, 21316}, {7280, 33965}, {10304, 66792}, {11749, 62069}, {12006, 68070}, {12017, 66805}, {12041, 14934}, {12052, 68084}, {12108, 68309}, {12121, 65086}, {12295, 21317}, {13391, 68074}, {14480, 15041}, {14508, 32609}, {14611, 51522}, {14891, 66818}, {15051, 36193}, {15111, 35495}, {15646, 47327}, {15688, 34312}, {15692, 66788}, {15693, 66786}, {15698, 66817}, {15710, 66820}, {15712, 18319}, {16111, 68472}, {16163, 16340}, {17502, 66789}, {17511, 38723}, {21269, 23515}, {32423, 55319}, {34128, 34150}, {34209, 38727}, {34577, 63708}, {36172, 38794}, {37950, 70569}, {37968, 62501}, {45694, 46045}, {54173, 66810}, {55610, 66807}, {62067, 66802}, {62100, 66791}, {66776, 67706}

X(72487) = midpoint of X(i) and X(j) for these {i,j}: {3, 38610}, {20, 66795}, {477, 38609}, {550, 3258}, {1511, 36164}, {3579, 66770}, {12041, 14934}, {12295, 21317}, {14611, 51522}, {16111, 68472}, {16163, 16340}, {37950, 70569}
X(72487) = reflection of X(i) in X(j) for these {i,j}: {21316, 6723}, {22104, 3530}, {68070, 12006}, {68084, 12052}, {68307, 3}, {68308, 3628}, {68309, 12108}
X(72487) = complement of X(66778)
X(72487) = reflection of X(i) in the line X(j)X(k) for these {i,j,k}: {61574, 140, 523}, {68555, 5, 523}
X(72487) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {3, 38610, 47084}, {550, 3154, 3258}, {22104, 40557, 55308}
X(72487) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 477, 38609}, {3, 38581, 38700}, {3, 38610, 16168}, {3, 38701, 38610}, {20, 57306, 66795}, {477, 38609, 16168}, {3523, 66773, 57305}, {5054, 66772, 66787}, {38609, 38610, 477}

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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.