ETC

X(72420) = X(2)X(9291)∩X(4)X(290)

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

Antreas Hatzipolakis and Ercole Suppa, euclid 9541.

X(72420) lies on these lines: {2, 9291}, {4, 290}, {20, 16089}, {69, 57677}, {76, 1093}, {107, 37465}, {194, 47739}, {253, 264}, {276, 3090}, {311, 59139}, {317, 42355}, {381, 42368}, {393, 2998}, {683, 6524}, {1235, 52661}, {1975, 15143}, {2052, 2996}, {5071, 55079}, {6331, 6337}, {6530, 44144}, {8795, 15077}, {11185, 62274}, {13450, 44146}, {14618, 53173}, {16081, 64983}, {18817, 59428}, {22456, 53783}, {27376, 42359}, {40680, 68535}, {52448, 62949}

X(72420) = polar conjugate of X(51336)
X(72420) = lies on all circumconics with perspector on the line {30476, 40887}
X(72420) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(4),X(1968)}, {A,B,C,X(69),X(59527)}, {A,B,C,X(253),X(290)}, {A,B,C,X(263),X(40951)}, {A,B,C,X(6526),X(57677)}}
X(72420) = pole of tripolar of X(51336) with respect to polar circle
X(72420) = pole of the line {16229, 62521} with respect to Steiner circumellipse
X(72420) = pole of the line {417, 10607} with respect to Wallace hyperbola
X(72420) = trilinear pole of line {X(30476), X(40887)}
X(72420) = barycentric product X(i)*X(j) for these (i, j): {264, 9308}, {290, 40887}, {823, 17893}, {1957, 1969}, {1958, 57806}, {1968, 18022}, {1975, 2052}, {6331, 16229}, {6528, 30476}, {9306, 18027}, {15143, 60199}, {17478, 57973}
X(72420) = barycentric quotient X(i)/X(j) for these (i, j): {4, 51336}, {92, 9255}, {158, 9258}, {264, 9289}, {393, 9292}, {1957, 48}, {1958, 255}, {1968, 184}, {1975, 394}, {2052, 9307}, {2451, 39201}, {2996, 60834}, {6528, 43188}, {9306, 577}, {9308, 3}, {15143, 3289}, {15352, 65837}, {16229, 647}, {17215, 4091}, {17478, 822}, {17893, 24018}, {22089, 32320}, {30476, 520}, {37199, 3167}, {40887, 511}, {59527, 6509}, {59561, 22401}, {60841, 60833}, {64983, 43727}
X(72420) = trilinear product X(i)*X(j) for these (i, j): {92, 9308}, {107, 17893}, {158, 1975}, {264, 1957}, {811, 16229}, {821, 59527}, {823, 30476}, {1821, 40887}, {1958, 2052}, {1968, 1969}, {2451, 57973}, {6528, 17478}, {9306, 57806}
X(72420) = trilinear quotient X(i)/X(j) for these (i, j): {48, 9308}, {92, 51336}, {158, 9292}, {184, 1957}, {255, 1975}, {264, 9255}, {520, 17893}, {577, 1958}, {810, 16229}, {820, 59527}, {822, 30476}, {1755, 40887}, {1968, 9247}, {1969, 9289}, {2052, 9258}, {9306, 52430}, {9307, 57806}, {17215, 23224}, {17478, 39201}, {43188, 57973}
X(72420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6528, 18027, 4}, {9291, 62576, 2}

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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.

ORTHOLOGIC TRIANGLES

Let ABC be a triangle, P, Q two points, A'B'C' the padal triangle of Q and S a point on the Euler line such that OS/OH= t: number.

Denote:

Ab, Ac = the orthogonal projections of A on BP, CP, resp.
Sa = same to S point of the triangle A'AbAc.
Similary Sb,Sc.

Cases the triangle ABC, SaSbSc are orthologic.

1. Let ABC be a triangle, P a point and A'B'C' the pedal triangle of a point Q.

Denote:

Ab, Ac = the orthogonal projections of A on BP, CP, resp.

Ga = the centroid of A'AbAc.
Similarly Gb, Gc

ABC, GaGbGc are circumorthologic.
ie The Orthologic center (ABC, GaGbGc) = X1 lies on the circumcircle of ABC.
The Orthologic center (GaGbGc, ABC) = X2 lies on the circumcircle of GaGbGc.

2. Let ABC be a triangle, P = I = X(1), A'B'C' the pedal triangle of a point Q and S a point on the Euler line such that OS/OH= t: number.

Denote:

Ab, Ac = the orthogonal projections of A on BI, CI, resp.

Sa = same to S point of the triangle A'AbAc.
Similarly Sb, Sc

ABC, SaSbSc are Orthologic.

3. Let ABC be a triangle, P, Q two isogonal conjugate points, A'B'C' the pedal triangle of Q and S a point on the Euler line such that OS/OH= t: number.

Denote:

Ab, Ac = the orthogonal projections of A on BP, CP, resp.

Sa = same to S point of the triangle A'AbAc.
Similarly Sb, Sc

ABC, SaSbSc are circumoerthologic. ie The Orthologic center (ABC, SaSbSc) = X1 lies on the circumcircle of ABC.
The Orthologic center (SaSbSc, ABC) = X2 lies on the circumcircle of SaSbSc.

EULER

X(72398) = 105TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 6*a^10 - 11*a^8*b^2 - 2*a^6*b^4 + 12*a^4*b^6 - 4*a^2*b^8 - b^10 - 11*a^8*c^2 + 38*a^6*b^2*c^2 - 21*a^4*b^4*c^2 - 9*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 21*a^4*b^2*c^4 + 26*a^2*b^4*c^4 - 2*b^6*c^4 + 12*a^4*c^6 - 9*a^2*b^2*c^6 - 2*b^4*c^6 - 4*a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(72398) = 5 X[3] - 3 X[16532], 3 X[3] - X[43893], 7 X[3] - 3 X[46451], X[23] - 4 X[62087], 3 X[140] - 2 X[403], 5 X[140] - 4 X[15350], 5 X[140] - 2 X[44267], 3 X[376] + X[35452], 4 X[403] - 3 X[11558], 5 X[403] - 6 X[15350], X[403] - 3 X[34152], 5 X[403] - 3 X[44267], 4 X[468] - 7 X[61784], 5 X[546] - 8 X[5159], 7 X[546] - 8 X[63838], 3 X[547] - 4 X[10257], 7 X[547] - 4 X[47310], 25 X[548] - 4 X[37899], 9 X[548] - 4 X[37931], 7 X[548] - 4 X[47335], 19 X[548] - 4 X[47342], 3 X[549] - X[52403], 5 X[550] + X[5189], 3 X[550] - X[13619], 2 X[858] + X[62151], X[1657] + 3 X[44450], X[2070] - 3 X[8703], 5 X[2071] - X[18403], 3 X[2071] - X[37938], 9 X[2071] - X[64890], 7 X[2072] - 3 X[65087], 5 X[3522] - X[5899], 7 X[3528] - 3 X[37922], 2 X[3530] - 3 X[37948], 3 X[3534] + X[46450], X[3627] - 3 X[65085], 7 X[3853] - 6 X[65087], 3 X[5066] - 2 X[44283], 7 X[5159] - 5 X[63838], 3 X[5189] + 5 X[13619], X[5189] - 5 X[18859], 3 X[7426] - 5 X[15646], 2 X[7426] - 5 X[34200], X[7464] + 2 X[44245], X[7574] + 2 X[62136], 5 X[10096] - 6 X[16532], 3 X[10096] - 2 X[43893], 7 X[10096] - 6 X[46451], 4 X[10151] - 5 X[61940], 7 X[10257] - 3 X[47310], 5 X[11558] - 8 X[15350], X[11558] - 4 X[34152], 5 X[11558] - 4 X[44267], X[11563] - 3 X[37948], 2 X[11799] - 5 X[61790], 3 X[12100] - 2 X[44234], 3 X[12101] - 4 X[23323], X[12103] + 2 X[37950], 5 X[12812] - 2 X[62288], X[13473] - 3 X[15122], 4 X[13473] - 3 X[62026], X[13619] + 3 X[18859], 3 X[14893] - 2 X[64891], 4 X[15122] - X[62026], 2 X[15350] - 5 X[34152], 2 X[15646] - 3 X[34200], 9 X[15688] - X[37949], 3 X[15690] - 2 X[44246], X[15690] + 2 X[54995], 5 X[15712] - 3 X[37943], 9 X[16532] - 5 X[43893], 7 X[16532] - 5 X[46451], 8 X[16976] - 7 X[61821], X[18325] - 4 X[61792], 3 X[18403] - 5 X[37938], 9 X[18403] - 5 X[64890], 2 X[18572] + X[62156], X[20063] - 13 X[62105], 5 X[22248] - 2 X[62344], 3 X[25338] - 4 X[37935], 7 X[25338] - 4 X[47338], 2 X[25338] - 5 X[62064], 5 X[30745] - 2 X[62013], 5 X[34152] - X[44267], X[35001] + 5 X[62104], 3 X[35489] - 7 X[62100], 5 X[37760] - 11 X[62062], 9 X[37899] - 25 X[37931], 7 X[37899] - 25 X[47335], 19 X[37899] - 25 X[47342], 5 X[37923] - 17 X[62084], X[37924] - 7 X[62091], X[37925] - 6 X[62089], 7 X[37931] - 9 X[47335], 19 X[37931] - 9 X[47342], 2 X[37935] - 3 X[37968], 7 X[37935] - 3 X[47338], 8 X[37935] - 15 X[62064], 3 X[37938] - X[64890], 3 X[37941] - 4 X[58190], X[37944] + 4 X[41981], X[37947] - 3 X[44280], 3 X[37955] - 2 X[44264], 3 X[37955] - 5 X[46853], 3 X[37956] - 11 X[62085], 5 X[37958] - 11 X[62079], 7 X[37968] - 2 X[47338], 4 X[37968] - 5 X[62064], 9 X[41982] - 8 X[47114], 3 X[41983] - 2 X[44282], 7 X[43893] - 9 X[46451], 2 X[44214] - 3 X[61782], X[44246] + 3 X[54995], 2 X[44264] - 5 X[46853], 4 X[44452] - 5 X[61810], 8 X[44911] - 9 X[47598], 4 X[44961] - 7 X[61821], 4 X[46031] - 5 X[48154], 4 X[47090] + X[58203], 4 X[47311] + 5 X[62138], 19 X[47335] - 7 X[47342], 4 X[47336] - 7 X[55862], 8 X[47338] - 35 X[62064], 7 X[50693] + X[60466], 5 X[60455] + 7 X[62134], 3 X[60462] + 5 X[62131], 13 X[62092] - X[62290], 3 X[13363] - 2 X[13446]

See Antreas Hatzipolakis and Peter Moses, euclid 9446.

X(72398) lies on these lines: {2, 3}, {74, 50708}, {477, 33639}, {930, 67735}, {1154, 17855}, {1291, 67797}, {1294, 13863}, {2693, 30248}, {2777, 46114}, {6799, 53934}, {13363, 13446}, {13391, 37853}, {13399, 32423}, {13445, 34153}, {14677, 43574}, {22115, 43391}, {29011, 67784}, {40111, 50434}, {53884, 67727}

X(72398) = midpoint of X(i) and X(j) for these {i,j}: {550, 18859}, {3153, 15704}, {13445, 34153}, {14677, 43574}, {16386, 37950}, {40111, 50434}
X(72398) = reflection of X(i) in X(j) for these {i,j}: {140, 34152}, {186, 33923}, {3853, 2072}, {10096, 3}, {11558, 140}, {11563, 3530}, {12103, 16386}, {25338, 37968}, {31726, 3628}, {44267, 15350}, {44961, 16976}, {47096, 22249}
X(72398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {403, 13473, 44226}, {427, 21284, 65154}, {3520, 13619, 403}, {3530, 50143, 140}, {5159, 7426, 6677}, {5189, 6636, 7426}, {11563, 37948, 3530}, {15690, 66718, 548}, {16387, 47311, 5159}

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X(72399) = 106TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 4*a^10 - 12*a^8*b^2 + 8*a^6*b^4 + 8*a^4*b^6 - 12*a^2*b^8 + 4*b^10 - 12*a^8*c^2 + 2*a^6*b^2*c^2 - 7*a^4*b^4*c^2 + 29*a^2*b^6*c^2 - 12*b^8*c^2 + 8*a^6*c^4 - 7*a^4*b^2*c^4 - 34*a^2*b^4*c^4 + 8*b^6*c^4 + 8*a^4*c^6 + 29*a^2*b^2*c^6 + 8*b^4*c^6 - 12*a^2*c^8 - 12*b^2*c^8 + 4*c^10 : :
X(72399) = X[2] - 9 X[37943], X[2] - 3 X[44282], 25 X[2] - 9 X[44450], 7 X[2] + 9 X[46451], 17 X[2] - 9 X[65085], 2 X[5] + X[12105], X[23] + 3 X[5055], 7 X[140] + 2 X[47338], 3 X[186] + X[3830], 5 X[381] - X[10296], X[381] + 3 X[37907], 3 X[403] - X[3845], 3 X[403] + X[44265], 4 X[468] - X[18571], 3 X[468] - X[18579], 5 X[468] - 2 X[22249], 2 X[468] + X[44961], 9 X[468] - X[47031], 13 X[468] - X[47308], 11 X[468] + X[47309], 7 X[468] + X[47310], 3 X[468] + X[47332], 5 X[468] - X[47333], 7 X[468] - X[47335], 5 X[468] + X[47336], 3 X[549] - X[54995], 5 X[632] + X[62344], X[858] - 3 X[15699], 5 X[1656] - X[10989], 5 X[1656] + X[37967], 3 X[2070] + 5 X[19709], 3 X[2071] - 7 X[15701], 3 X[2072] + X[47313], 3 X[2072] - 5 X[61910], 7 X[3090] + X[37901], 3 X[3153] - 11 X[61932], 3 X[3524] + X[18325], X[3534] - 3 X[15646], 3 X[3545] - X[18572], 3 X[3545] + 5 X[37760], 2 X[3628] + X[16619], 5 X[3843] + 7 X[37957], 7 X[3851] + 5 X[37953], X[3853] + 2 X[37934], 3 X[5054] - X[37950], X[5066] + 3 X[10096], X[5066] - 6 X[37942], 2 X[5066] - 3 X[46031], 7 X[5066] - 6 X[63838], 11 X[5070] + X[37946], 5 X[5071] - X[7574], 5 X[5071] + 3 X[37909], 2 X[5159] - 3 X[47599], X[5189] - 9 X[61899], 3 X[5899] + 13 X[61901], X[7464] - 5 X[15694], X[7574] + 3 X[37909], 5 X[7575] + X[10296], X[7575] - 3 X[37907], X[8703] + 3 X[11563], 5 X[8703] - 3 X[16386], 2 X[8703] - 3 X[37968], X[8703] - 3 X[44214], X[8703] - 6 X[44900], X[10096] + 2 X[37942], 2 X[10096] + X[46031], 7 X[10096] + 2 X[63838], 2 X[10109] + X[37904], 3 X[10151] - 2 X[61997], 3 X[10257] - 4 X[11540], X[10296] + 15 X[37907], X[10297] + 2 X[44264], X[11001] + 3 X[31726], X[11558] + 2 X[16531], 3 X[11558] + X[62138], 5 X[11563] + X[16386], 2 X[11563] + X[37968], X[11563] + 2 X[44900], 3 X[11799] + X[54995], 2 X[11812] - 3 X[44452], X[12100] - 3 X[44234], 4 X[12811] - X[47339], 3 X[13619] + 5 X[62007], 3 X[14269] + 5 X[37958], 3 X[14892] + 4 X[47316], 2 X[15350] + X[37971], 6 X[15350] - X[47311], 3 X[15350] - 2 X[61896], X[15681] - 5 X[37952], X[15682] - 3 X[44283], X[15685] - 9 X[37955], 5 X[15693] - 3 X[34152], 5 X[15695] - 9 X[37941], 7 X[15703] + X[37924], 5 X[15713] + 3 X[43893], 2 X[16386] - 5 X[37968], X[16386] - 5 X[44214], X[16386] - 10 X[44900], 6 X[16531] - X[62138], 9 X[16532] - X[19710], 3 X[16532] - X[44280], X[18323] - 3 X[23046], 3 X[18403] - 7 X[41106], 3 X[18571] - 4 X[18579], 5 X[18571] - 8 X[22249], X[18571] + 2 X[44961], 9 X[18571] - 4 X[47031], 13 X[18571] - 4 X[47308], 11 X[18571] + 4 X[47309], 7 X[18571] + 4 X[47310], 3 X[18571] + 4 X[47332], 5 X[18571] - 4 X[47333], X[18571] + 4 X[47334], 7 X[18571] - 4 X[47335], 5 X[18571] + 4 X[47336], X[18572] + 5 X[37760], 5 X[18579] - 6 X[22249], 2 X[18579] + 3 X[44961], 3 X[18579] - X[47031], 13 X[18579] - 3 X[47308], 11 X[18579] + 3 X[47309], 7 X[18579] + 3 X[47310], 5 X[18579] - 3 X[47333], X[18579] + 3 X[47334], 7 X[18579] - 3 X[47335], 5 X[18579] + 3 X[47336], 3 X[18859] - 11 X[61843], 5 X[19708] + 3 X[52403], X[19710] - 3 X[44280], X[20063] + 15 X[61906], 5 X[22248] + 3 X[41987], 4 X[22249] + 5 X[44961], 18 X[22249] - 5 X[47031], 26 X[22249] - 5 X[47308], 22 X[22249] + 5 X[47309], 14 X[22249] + 5 X[47310], 6 X[22249] + 5 X[47332], 2 X[22249] + 5 X[47334], 14 X[22249] - 5 X[47335], 2 X[22249] + X[47336], 3 X[23323] - 4 X[61960], X[25338] + 2 X[68319], 5 X[30745] - 9 X[61887], 7 X[33699] - 9 X[65087], X[35001] - 9 X[61864], 4 X[35018] + X[47312], 4 X[35018] - X[47341], 3 X[35452] - 19 X[61857], 9 X[35489] + 7 X[62009], 2 X[37897] + 3 X[47478], X[37899] + 6 X[45757], X[37900] + 9 X[61909], 4 X[37911] - 3 X[47598], 9 X[37922] + 7 X[61974], 5 X[37923] + 11 X[61925], 3 X[37925] + 17 X[61893], 3 X[37931] + 2 X[62010], 6 X[37935] + X[62022], 3 X[37936] + 7 X[61920], 3 X[37938] - X[47314], 3 X[37938] - 7 X[61898], 9 X[37940] + 11 X[61950], 4 X[37942] - X[46031], 7 X[37942] - X[63838], 9 X[37943] + X[44266], 3 X[37943] - X[44282], 25 X[37943] - X[44450], 7 X[37943] + X[46451], 17 X[37943] - X[65085], 3 X[37944] - 23 X[61862], 3 X[37947] + 11 X[61908], 9 X[37948] - 13 X[61797], X[37968] - 4 X[44900], 3 X[37971] + X[47311], 3 X[37971] + 4 X[61896], 3 X[38335] + X[56369], 3 X[44246] - X[62154], X[44266] + 3 X[44282], 25 X[44266] + 9 X[44450], 7 X[44266] - 9 X[46451], 17 X[44266] + 9 X[65085], 25 X[44282] - 3 X[44450], 7 X[44282] + 3 X[46451], 17 X[44282] - 3 X[65085], 7 X[44450] + 25 X[46451], 17 X[44450] - 25 X[65085], 9 X[44961] + 2 X[47031], 13 X[44961] + 2 X[47308], 11 X[44961] - 2 X[47309], 7 X[44961] - 2 X[47310], 3 X[44961] - 2 X[47332], 5 X[44961] + 2 X[47333], 7 X[44961] + 2 X[47335], 5 X[44961] - 2 X[47336], 7 X[46031] - 4 X[63838], 3 X[46450] - 19 X[61913], 17 X[46451] + 7 X[65085], 13 X[47031] - 9 X[47308], 11 X[47031] + 9 X[47309], 7 X[47031] + 9 X[47310], X[47031] + 3 X[47332], 5 X[47031] - 9 X[47333], X[47031] + 9 X[47334], 7 X[47031] - 9 X[47335], 5 X[47031] + 9 X[47336], 3 X[47096] + 7 X[61851], 11 X[47308] + 13 X[47309], 7 X[47308] + 13 X[47310], 3 X[47308] + 13 X[47332], 5 X[47308] - 13 X[47333], X[47308] + 13 X[47334], 7 X[47308] - 13 X[47335], 5 X[47308] + 13 X[47336], 7 X[47309] - 11 X[47310], 3 X[47309] - 11 X[47332], 5 X[47309] + 11 X[47333], X[47309] - 11 X[47334], 7 X[47309] + 11 X[47335], 5 X[47309] - 11 X[47336], 3 X[47310] - 7 X[47332], 5 X[47310] + 7 X[47333], X[47310] - 7 X[47334], 5 X[47310] - 7 X[47336], X[47311] - 4 X[61896], X[47313] + 5 X[61910], X[47314] - 7 X[61898], 5 X[47332] + 3 X[47333], X[47332] - 3 X[47334], 7 X[47332] + 3 X[47335], 5 X[47332] - 3 X[47336], X[47333] + 5 X[47334], 7 X[47333] - 5 X[47335], 7 X[47334] + X[47335], 5 X[47334] - X[47336], 5 X[47335] + 7 X[47336], X[47340] + 4 X[67236], X[47342] + 4 X[61922], 7 X[55856] - X[62332], 3 X[57584] - 5 X[61998], 5 X[60455] - 21 X[61897], 15 X[61882] + X[62290], 7 X[62000] - 3 X[64890], X[62043] - 3 X[64891], X[110] + 3 X[15362], 3 X[5215] - X[38611], X[9158] + 3 X[57305], X[11179] - 5 X[47453], X[11801] + 2 X[15448], 3 X[14643] + X[15360], 2 X[15088] + X[32237], X[20423] + 3 X[47450], X[21850] + 5 X[47452], X[34315] + 3 X[59403], X[34316] + 3 X[59404], 3 X[47455] - X[50979], X[47471] + 3 X[47562], X[50955] + 3 X[52238]

See Antreas Hatzipolakis and Peter Moses, euclid 9446.

X(72399) lies on these lines: {2, 3}, {110, 15362}, {113, 15361}, {524, 10272}, {952, 47495}, {3564, 47544}, {5215, 38611}, {5844, 47488}, {9158, 57305}, {11178, 32217}, {11179, 47453}, {11645, 20304}, {11649, 13364}, {11801, 15448}, {12900, 19924}, {14643, 15360}, {15088, 32237}, {16328, 18487}, {20423, 47450}, {21850, 47452}, {32423, 35266}, {32515, 46986}, {34315, 59403}, {34316, 59404}, {34380, 47473}, {43291, 47169}, {43656, 53950}, {44204, 47219}, {44569, 46817}, {45969, 61606}, {47455, 50979}, {47471, 47562}, {47556, 47581}, {50955, 52238}, {61572, 62508}, {61619, 63124}

X(72399) = midpoint of X(i) and X(j) for these {i,j}: {2, 44266}, {5, 7426}, {113, 15361}, {376, 44267}, {381, 7575}, {468, 47334}, {547, 25338}, {549, 11799}, {3845, 44265}, {10295, 15687}, {10989, 37967}, {11178, 32217}, {11563, 44214}, {11737, 44264}, {15686, 62288}, {16619, 47097}, {18579, 47332}, {44204, 47219}, {44569, 46817}, {47310, 47335}, {47312, 47341}, {47333, 47336}, {47556, 47581}
X(72399) = reflection of X(i) in X(j) for these {i,j}: {547, 68319}, {10297, 11737}, {12105, 7426}, {14893, 37984}, {15122, 10124}, {37968, 44214}, {44214, 44900}, {44961, 47334}, {47097, 3628}, {47333, 22249}, {62139, 66595}
X(72399) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {381, 37907, 7575}, {403, 44265, 3845}, {403, 66725, 37984}, {468, 44961, 18571}, {468, 47332, 18579}, {468, 47336, 22249}, {5071, 37909, 7574}, {10096, 37942, 46031}, {10096, 44233, 25338}, {10109, 66529, 5066}, {10296, 10298, 16386}, {11563, 44900, 37968}, {13626, 13627, 381}, {14002, 37907, 7426}, {18579, 47334, 47332}, {25338, 44234, 25337}, {34330, 62961, 14893}, {44233, 68319, 46031}, {44266, 44282, 2}, {57322, 57323, 61924}

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TWO PROBLEMS CONNECTED

Francisco Javier García Capitán, Two problems [by Antreas Hatzipolakis] connected.

FGJC
geogebra file

z

X(72407) = X(13)X(42788)∩X(14)X(1506)

Barycentrics    8 a^16 - 84 a^14 b^2 + 375 a^12 b^4 - 908 a^10 b^6 + 1288 a^8 b^8 - 1086 a^6 b^10 + 528 a^4 b^12 - 136 a^2 b^14 + 15 b^16 - 84 a^14 c^2 + 594 a^12 b^2 c^2 - 1492 a^10 b^4 c^2 + 1258 a^8 b^6 c^2 + 552 a^6 b^8 c^2 - 1476 a^4 b^10 c^2 + 766 a^2 b^12 c^2 - 130 b^14 c^2 + 375 a^12 c^4 - 1492 a^10 b^2 c^4 + 1076 a^8 b^4 c^4 + 1002 a^6 b^6 c^4 + 391 a^4 b^8 c^4 - 1492 a^2 b^10 c^4 + 487 b^12 c^4 - 908 a^10 c^6 + 1258 a^8 b^2 c^6 + 1002 a^6 b^4 c^6 + 1082 a^4 b^6 c^6 + 862 a^2 b^8 c^6 - 1018 b^10 c^6 + 1288 a^8 c^8 + 552 a^6 b^2 c^8 + 391 a^4 b^4 c^8 + 862 a^2 b^6 c^8 + 1292 b^8 c^8 - 1086 a^6 c^10 - 1476 a^4 b^2 c^10 - 1492 a^2 b^4 c^10 - 1018 b^6 c^10 + 528 a^4 c^12 + 766 a^2 b^2 c^12 + 487 b^4 c^12 - 136 a^2 c^14 - 130 b^2 c^14 + 15 c^16 - 4 a^14 T + 28 a^12 b^2 T - 86 a^10 b^4 T + 138 a^8 b^6 T - 134 a^6 b^8 T + 78 a^4 b^10 T - 22 a^2 b^12 T + 2 b^14 T + 28 a^12 c^2 T - 112 a^10 b^2 c^2 T + 142 a^8 b^4 c^2 T - 48 a^6 b^6 c^2 T - 58 a^4 b^8 c^2 T + 56 a^2 b^10 c^2 T - 6 b^12 c^2 T - 86 a^10 c^4 T + 142 a^8 b^2 c^4 T - 112 a^6 b^4 c^4 T + 56 a^4 b^6 c^4 T - 84 a^2 b^8 c^4 T + 8 b^10 c^4 T + 138 a^8 c^6 T - 48 a^6 b^2 c^6 T + 56 a^4 b^4 c^6 T + 96 a^2 b^6 c^6 T - 4 b^8 c^6 T - 134 a^6 c^8 T - 58 a^4 b^2 c^8 T - 84 a^2 b^4 c^8 T - 4 b^6 c^8 T + 78 a^4 c^10 T + 56 a^2 b^2 c^10 T + 8 b^4 c^10 T - 22 a^2 c^12 T - 6 b^2 c^12 T + 2 c^14 T : : where T = Sqrt[3] S

Benjamin Lee Warren and Francisco Javier García Capitán, euclid 9479.

X(72407) lies on these lines: {5, 11602}, {13, 42788}, {14, 1506}, {17, 5615}, {18, 59403}, {76, 16966}, {83, 6671}, {5487, 35689}, {6115, 11606}, {10187, 25555}, {11122, 37832}, {11272, 43539}, {12817, 52649}, {16242, 62877}, {16964, 54861}

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X(72408) = X(13)X(1506)∩X(14)X(42788)

Barycentrics    8 a^16-84 a^14 b^2+375 a^12 b^4-908 a^10 b^6+1288 a^8 b^8-1086 a^6 b^10+528 a^4 b^12-136 a^2 b^14+15 b^16-84 a^14 c^2+594 a^12 b^2 c^2-1492 a^10 b^4 c^2+1258 a^8 b^6 c^2+552 a^6 b^8 c^2-1476 a^4 b^10 c^2+766 a^2 b^12 c^2-130 b^14 c^2+375 a^12 c^4-1492 a^10 b^2 c^4+1076 a^8 b^4 c^4+1002 a^6 b^6 c^4+391 a^4 b^8 c^4-1492 a^2 b^10 c^4+487 b^12 c^4-908 a^10 c^6+1258 a^8 b^2 c^6+1002 a^6 b^4 c^6+1082 a^4 b^6 c^6+862 a^2 b^8 c^6-1018 b^10 c^6+1288 a^8 c^8+552 a^6 b^2 c^8+391 a^4 b^4 c^8+862 a^2 b^6 c^8+1292 b^8 c^8-1086 a^6 c^10-1476 a^4 b^2 c^10-1492 a^2 b^4 c^10-1018 b^6 c^10+528 a^4 c^12+766 a^2 b^2 c^12+487 b^4 c^12-136 a^2 c^14-130 b^2 c^14+15 c^16+4 a^14 T-28 a^12 b^2 T+86 a^10 b^4 T-138 a^8 b^6 T+134 a^6 b^8 T-78 a^4 b^10 T+22 a^2 b^12 T-2 b^14 T-28 a^12 c^2 T+112 a^10 b^2 c^2 T-142 a^8 b^4 c^2 T+48 a^6 b^6 c^2 T+58 a^4 b^8 c^2 T-56 a^2 b^10 c^2 T+6 b^12 c^2 T+86 a^10 c^4 T-142 a^8 b^2 c^4 T+112 a^6 b^4 c^4 T-56 a^4 b^6 c^4 T+84 a^2 b^8 c^4 T-8 b^10 c^4 T-138 a^8 c^6 T+48 a^6 b^2 c^6 T-56 a^4 b^4 c^6 T-96 a^2 b^6 c^6 T+4 b^8 c^6 T+134 a^6 c^8 T+58 a^4 b^2 c^8 T+84 a^2 b^4 c^8 T+4 b^6 c^8 T-78 a^4 c^10 T-56 a^2 b^2 c^10 T-8 b^4 c^10 T+22 a^2 c^12 T+6 b^2 c^12 T-2 c^14 T : : where T = Sqrt[3] S

Benjamin Lee Warren and Francisco Javier García Capitán, euclid 9479.

X(72408) lies on these lines: {5, 11603}, {13, 1506}, {14, 42788}, {17, 59404}, {18, 5611}, {76, 16967}, {83, 6672}, {5488, 35688}, {6114, 11606}, {10188, 25555}, {11121, 37835}, {11272, 43538}, {12816, 44289}, {16241, 62876}, {16965, 54860}

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ETC

X(5459)
Let ABC be a triangle, let A', B', C' be the midpoints of BC, CA, AB. Let L_a be the perpendicular through A' to the line X(618)A'. Define L_b, L_c cyclically. Then X(5459) is the center of the equilateral triangle A''B''C'' bounded by L_a, L_b, L_c. The circumcircle of A''B''C'' passes through X(14081) and X(32552) and has squared radius (3 sqrt(3) S^3 + 9 S^2 SW + 3 sqrt(3) S SW^2 + SW^3)/(9 (3 S^2 + 2 sqrt(3) S SW + SW^2)). This circle (A''B''C'') is here named 1st Suppa circle. The 2nd Suppa circle is defined at X(5460)
(Euclid 8675, August 28, 2025)

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X(5960)
Let ABC be a triangle, let A', B', C' be the midpoints of BC, CA, AB. Let L_a be the perpendicular through A' to the line X(619)A'. Define L_b, L_c cyclically. Then X(5460) is the center of the equilateral triangle A''B''C'' bounded by L_a, L_b, L_c. The circumcircle of A''B''C'' passes through X(14082) and X(32553)and has squared radius (-3 sqrt(3) S^3 + 9 S^2 SW - 3 sqrt(3) S SW^2 + SW^3)/(9 (3 S^2 - 2 sqrt(3) S SW + SW^2)). The circle (A''B''C'') is here named 2nd Suppa circle. The 1st Suppa circle is defined at X(5459)
(Euclid 8675, August 28, 2025)