ETC

X(72801) = X(10)X(79)∩X(35)X(502)

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

Francisco Javier García Capitán, euclid 9871.

X(72801) liea on circumconics with center X(i) for i in {4092, 21709} and these lines: {10, 79}, {35, 502}, {42, 1989}, {171, 13486}, {210, 8013}, {226, 21054}, {3214, 56843}, {3701, 6757}, {4518, 30690}, {4705, 15475}, {6186, 7110}, {6742, 14844}, {7100, 59305}, {9140, 33097}, {20488, 69298}, {20499, 33078}, {21077, 21682}, {21089, 32936}, {21674, 52382}, {21686, 52390}, {21717, 63171}, {21729, 71630}, {23901, 33081}, {23927, 61358}, {23930, 69300}, {27577, 56847}, {36815, 52449}, {41504, 69545}, {43682, 70316}, {56191, 56844}

X(72801) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(10),X(6535)}, {A,B,C,X(12),X(210)}, {A,B,C,X(37),X(21873)}, {A,B,C,X(42),X(4705)}, {A,B,C,X(86),X(4683)}, {A,B,C,X(226),X(4024)}, {A,B,C,X(321),X(6543)}, {A,B,C,X(430),X(6175)}, {A,B,C,X(502),X(6058)}, {A,B,C,X(523),X(55925)}}
X(72801) = pole of tripolar of X(16755) with respect to the dual conic of Wallace hyperbola
X(72801) = barycentric product X(i)*X(j) for these (i, j): {10, 8818}, {12, 7110}, {37, 6757}, {79, 594}, {210, 43682}, {756, 30690}, {1089, 2160}, {1500, 20565}, {1577, 56193}, {1826, 52388}, {2166, 4053}, {2171, 52344}, {2321, 52382}, {3695, 64834}, {3952, 55236}, {4024, 6742}, {4705, 15455}, {6057, 52374}, {6186, 28654}, {6358, 7073}, {6535, 52393}, {7140, 52381}, {8013, 60139}, {21675, 57710}, {30713, 69470}, {43082, 68819}, {53008, 63171}, {55209, 58289}
X(72801) = barycentric quotient X(i)/X(j) for these (i, j): {10, 34016}, {12, 17095}, {37, 56934}, {42, 40214}, {79, 1509}, {181, 2003}, {210, 56440}, {213, 17104}, {523, 16755}, {594, 319}, {756, 3219}, {762, 3678}, {872, 2174}, {1089, 33939}, {1334, 35193}, {1500, 35}, {1962, 17190}, {2160, 757}, {2171, 1442}, {2643, 7202}, {3124, 53542}, {3952, 55235}, {4024, 4467}, {4036, 18160}, {4079, 2605}, {4705, 14838}, {6057, 42033}, {6186, 593}, {6358, 52421}, {6535, 3969}, {6742, 4610}, {6757, 274}, {7064, 52405}, {7073, 2185}, {7110, 261}, {7140, 52412}, {8013, 3578}, {8029, 21141}, {8606, 65568}, {8736, 7282}, {8818, 86}, {15455, 4623}, {21043, 8287}, {21794, 7279}, {21816, 3647}, {21824, 7266}, {21833, 2611}, {30690, 873}, {43682, 57785}, {52344, 52379}, {52374, 552}, {52375, 763}, {52382, 1434}, {52388, 17206}, {52393, 6628}, {55236, 7192}, {56193, 662}, {58289, 55210}, {59179, 30581}, {69470, 1412}
X(72801) = trilinear product X(i)*X(j) for these (i, j): {12, 7073}, {37, 8818}, {42, 6757}, {79, 756}, {181, 52344}, {210, 52382}, {523, 56193}, {594, 2160}, {762, 52393}, {872, 20565}, {1018, 55236}, {1089, 6186}, {1334, 43682}, {1500, 30690}, {1824, 52388}, {1989, 4053}, {2171, 7110}, {3701, 69470}, {3949, 64834}, {4079, 15455}, {4705, 6742}, {6057, 52372}, {6535, 52375}, {6538, 59179}, {7100, 7140}, {8013, 57419}, {8606, 56285}, {11060, 61410}, {21816, 60139}, {52390, 53008}
X(72801) = trilinear quotient X(i)/X(j) for these (i, j): {10, 56934}, {12, 1442}, {35, 756}, {37, 40214}, {42, 17104}, {60, 7073}, {79, 757}, {81, 8818}, {86, 6757}, {110, 56193}, {115, 7202}, {181, 1399}, {210, 35193}, {261, 52344}, {319, 1089}, {321, 34016}, {323, 4053}, {593, 2160}, {594, 3219}, {763, 52393}, {849, 6186}, {873, 20565}, {1014, 52382}, {1019, 55236}, {1213, 17190}, {1334, 35192}, {1408, 69470}, {1434, 43682}, {1444, 52388}, {1500, 2174}, {1509, 30690}, {1577, 16755}, {2003, 2171}, {2185, 7110}, {2321, 56440}, {2605, 4705}, {2611, 21043}, {2643, 53542}, {3647, 8013}, {3678, 6535}, {3690, 52408}, {4024, 14838}, {4033, 55235}, {4036, 4467}, {4092, 53524}, {4420, 6057}, {4610, 15455}, {6198, 7140}, {6358, 17095}, {6538, 59140}, {6742, 52935}, {7186, 7237}, {7266, 21054}, {7282, 56285}, {7341, 52372}, {7799, 61410}, {11107, 53008}, {16585, 21675}, {16718, 21018}, {17454, 21816}, {18160, 52623}, {20982, 21833}, {23226, 55230}, {28654, 33939}, {30593, 52569}, {34388, 52421}, {52558, 57419}
X(72801) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7073, 56193, 42}

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X(72802) = X(51)X(21354)∩X(418)X(8612)

Barycentrics    a^2*(a^2-b^2-c^2)*(a^2*b^2-b^4+a^2*c^2+2*b^2*c^2-c^4)^2*(a^8*b^4-4*a^6*b^6+6*a^4*b^8-4*a^2*b^10+b^12+a^10*c^2-2*a^6*b^4*c^2-4*a^4*b^6*c^2+9*a^2*b^8*c^2-4*b^10*c^2-4*a^8*c^4+2*a^4*b^4*c^4-4*a^2*b^6*c^4+6*b^8*c^4+6*a^6*c^6-2*a^2*b^4*c^6-4*b^6*c^6-4*a^4*c^8+b^4*c^8+a^2*c^10)*(a^10*b^2-4*a^8*b^4+6*a^6*b^6-4*a^4*b^8+a^2*b^10+a^8*c^4-2*a^6*b^2*c^4+2*a^4*b^4*c^4-2*a^2*b^6*c^4+b^8*c^4-4*a^6*c^6-4*a^4*b^2*c^6-4*a^2*b^4*c^6-4*b^6*c^6+6*a^4*c^8+9*a^2*b^2*c^8+6*b^4*c^8-4*a^2*c^10-4*b^2*c^10+c^12) : :

Francisco Javier García Capitán, euclid 9871.

X(72802) lies on these lines: {51, 21354}, {216, 34985}, {418, 8612}, {467, 1972}, {2055, 3484}

Χ(72802) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(216),X(324)}, {A,B,C,X(418),X(23607)}, {A,B,C,X(2055),X(33664)}}
Χ(72802) = barycentric quotient X(i)/X(j) for these (i, j): {46394, 2055}, {61378, 8613}, {62260, 56298}
Χ(72802) = trilinear quotient X(i)/X(j) for these (i, j): {56298, 62259}

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FOUR POINTS ON THE EULER LINE

Problem by Antreas Hatzipolakis
Solution by Francisco Javier García Capitán
FOUR POINTS ON THE EULER LINE

ETC LISTINGS

X(72698)

X(72699)

X(72700)

X(72701)

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?

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.
.