ETC1

X(73097) = X(5)X(51)∩X(30)X(5889)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(2*a^4 - 4*a^2*b^2 + 2*b^4 - 4*a^2*c^2 + b^2*c^2 + 2*c^4) : :
X(73097) = 3 X[2] - 4 X[16881], 5 X[3] - 3 X[62188], 5 X[3] - 7 X[67924], 3 X[62188] - 7 X[67924], 5 X[4] - 9 X[16981], 10 X[14449] - 9 X[16981], 5 X[5] - 6 X[51], 3 X[5] - 4 X[143], 3 X[5] - 2 X[5562], 7 X[5] - 6 X[5891], 7 X[5] - 8 X[10095], 5 X[5] - 4 X[11591], 11 X[5] - 12 X[13364], 9 X[5] - 8 X[14128], X[5] + 2 X[14531], 17 X[5] - 18 X[14845], 15 X[5] - 16 X[18874], 21 X[5] - 22 X[27355], 13 X[5] - 16 X[58533], 3 X[51] - 5 X[52], 9 X[51] - 10 X[143], 9 X[51] - 5 X[5562], 7 X[51] - 5 X[5891], 21 X[51] - 20 X[10095], 3 X[51] - 2 X[11591], 11 X[51] - 10 X[13364], 27 X[51] - 20 X[14128], 3 X[51] + 5 X[14531], 17 X[51] - 15 X[14845], 9 X[51] - 8 X[18874], 63 X[51] - 55 X[27355], 39 X[51] - 40 X[58533], 3 X[52] - 2 X[143], 3 X[52] - X[5562], 7 X[52] - 3 X[5891], 7 X[52] - 4 X[10095], 5 X[52] - 2 X[11591], 11 X[52] - 6 X[13364], 9 X[52] - 4 X[14128], 17 X[52] - 9 X[14845], 15 X[52] - 8 X[18874], 21 X[52] - 11 X[27355], 13 X[52] - 8 X[58533], 14 X[143] - 9 X[5891] and many others

Antreas Hatzipolakis and Peter Moses, euclid 10389.

X(73097) lies on these lines: {2, 16881}, {3, 1199}, {4, 14449}, {5, 51}, {24, 40111}, {26, 12160}, {30, 5889}, {49, 12107}, {68, 31815}, {140, 568}, {155, 37440}, {156, 37936}, {185, 13391}, {195, 7488}, {323, 45735}, {381, 31834}, {382, 62187}, {389, 549}, {397, 36979}, {398, 36981}, {511, 550}, {524, 38322}, {546, 3060}, {547, 11444}, {548, 5890}, {567, 7691}, {631, 54048}, {632, 1216}, {1112, 44960}, {1147, 7575}, {1204, 37950}, {1263, 27361}, {1351, 7526}, {1493, 18475}, {1531, 18555}, {1656, 66748}, {1658, 1993}, {1986, 34153}, {2070, 12316}, {2979, 3530}, {3090, 13321}, {3091, 13451}, {3357, 55720}, {3518, 50461}, {3519, 41171}, {3520, 32608}, {3526, 44324}, {3564, 9973}, {3567, 3628}, {3580, 10224}, {3581, 15331}, {3627, 10263}, {3819, 61853}, {3845, 5446}, {3850, 11459}, {3851, 11002}, {3853, 12111}, {3857, 10110}, {3858, 5907}, {3859, 66756}, {3861, 18435}, {3917, 12006}, {5066, 9781}, {5073, 64025}, {5462, 15067}, {5640, 35018}, {5663, 13421}, {5892, 15606}, {5899, 43605}, {5943, 61900}, {5944, 34986}, {5965, 45286}, {6000, 62041}, {6242, 7576}, {6515, 18569}, {6636, 43845}, {6662, 47301}, {6759, 37947}, {7502, 12161}, {7512, 15087}, {7514, 37493}, {7516, 11432}, {7525, 7592}, {7530, 12164}, {7556, 9704}, {7731, 16659}, {7999, 16239}, {8115, 20479}, {8116, 20478}, {8548, 11477}, {8703, 10625}, {8905, 14072}, {9729, 44682}, {9730, 10627}, {10124, 15028}, {10170, 61907}, {10226, 37495}, {10264, 23335}, {10574, 13340}, {10575, 62159}, {10602, 55724}, {10628, 41362}, {10982, 64105}, {11264, 11750}, {11381, 33699}, {11439, 12101}, {11451, 61894}, {11465, 61877}, {11539, 32142}, {11563, 22660}, {11793, 15026}, {11801, 12219}, {11803, 22815}, {12002, 46847}, {12022, 32339}, {12061, 64802}, {12085, 44456}, {12102, 15305}, {12103, 64050}, {12105, 26882}, {12108, 15045}, {12162, 15687}, {12225, 45970}, {12280, 14516}, {12290, 62034}, {12307, 14627}, {12325, 66766}, {12811, 15056}, {13142, 52070}, {13346, 34152}, {13348, 45759}, {13363, 55859}, {13369, 31819}, {13406, 66727}, {13470, 61713}, {13490, 31831}, {13491, 62155}, {13564, 15032}, {13861, 58891}, {14641, 44903}, {14677, 67912}, {14791, 15134}, {14855, 62126}, {14915, 62047}, {15012, 61802}, {15024, 48154}, {15030, 61976}, {15068, 31860}, {15072, 62144}, {15644, 46853}, {15646, 34114}, {15711, 17704}, {15713, 16226}, {15717, 54047}, {15720, 33884}, {15800, 58922}, {15806, 68085}, {16223, 22251}, {16266, 35602}, {16836, 61789}, {16880, 44879}, {16980, 61245}, {16982, 45187}, {17714, 18445}, {17800, 67925}, {18356, 31723}, {18438, 61624}, {18439, 62026}, {18567, 50435}, {18570, 36747}, {19710, 46850}, {20791, 58190}, {21849, 38071}, {22051, 32338}, {23046, 45958}, {23237, 68069}, {25147, 68063}, {31732, 34773}, {31738, 38028}, {31757, 38034}, {31760, 38042}, {31810, 65376}, {31833, 34380}, {32046, 37478}, {32139, 33586}, {32205, 61885}, {32269, 61608}, {33332, 67926}, {34200, 66606}, {34864, 54202}, {36987, 62106}, {37498, 68681}, {37705, 72908}, {38229, 39806}, {39522, 63682}, {40280, 61792}, {41725, 44668}, {43574, 43615}, {43575, 67337}, {43895, 64757}, {45780, 66762}, {46849, 62001}, {47582, 64472}, {50006, 72725}, {50136, 67883}, {52093, 62141}, {54041, 61790}, {54044, 61785}, {55718, 64180}, {58470, 61916}, {61136, 62100}, {61544, 61724}, {61753, 64095}, {61869, 72922}, {62104, 63414}, {62121, 66747}

X(73097) = midpoint of X(i) and X(j) for these {i,j}: {52, 14531}, {5073, 64025}, {5889, 6243}, {34783, 64051}
X(73097) = reflection of X(i) in X(j) for these {i,j}: {4, 14449}, {5, 52}, {550, 6102}, {1216, 16625}, {3627, 10263}, {5562, 143}, {5876, 5446}, {6101, 389}, {8703, 14831}, {10625, 13630}, {11412, 140}, {11750, 11264}, {12111, 3853}, {12162, 68084}, {12219, 11801}, {12225, 45970}, {12290, 62034}, {13369, 31819}, {14677, 67912}, {15687, 21969}, {15704, 185}, {18436, 546}, {18438, 61624}, {18439, 62026}, {22815, 11803}, {32338, 22051}, {34153, 1986}, {34773, 31732}, {37484, 548}, {37705, 72908}, {45186, 13421}, {45187, 45959}, {45731, 32358}, {45959, 16982}, {52070, 13142}, {61245, 16980}, {62036, 45186}, {62155, 13491}, {62159, 10575}, {64050, 12103}
X(73097) = X(72485)-Ceva conjugate of X(5)
X(73097) = X(2616)-isoconjugate of X(67739)
X(73097) = crosssum of X(3) and X(63652)
X(73097) = barycentric product X(343)*X(44879)
X(73097) = barycentric quotient X(i)/X(j) for these {i,j}: {1625, 67739}, {44879, 275}
X(73097) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 31802, 20424}, {51, 11591, 5}, {52, 5562, 143}, {68, 31815, 44288}, {143, 5562, 5}, {143, 11591, 18874}, {143, 18874, 51}, {389, 6101, 549}, {550, 6102, 45956}, {568, 11412, 140}, {1216, 5946, 632}, {1216, 16625, 5946}, {2070, 12316, 56292}, {2979, 37481, 3530}, {3060, 18436, 546}, {3090, 13321, 58531}, {3567, 23039, 3628}, {3581, 34148, 15331}, {3917, 12006, 14869}, {5446, 5876, 3845}, {5462, 15067, 55856}, {5562, 27355, 5891}, {5889, 64051, 34783}, {5890, 37484, 548}, {5891, 10095, 5}, {6243, 34783, 64051}, {7592, 37494, 7525}, {9729, 54042, 44682}, {9730, 10627, 15712}, {10110, 15060, 3857}, {10574, 13340, 33923}, {10625, 13630, 8703}, {10625, 14831, 13630}, {11793, 15026, 15699}, {12161, 17834, 7502}, {12162, 21969, 68084}, {12162, 68084, 15687}, {12307, 14627, 35921}, {15030, 67867, 61976}, {16226, 72921, 15713}, {16266, 37489, 37814}, {16982, 45187, 61988}, {31802, 64066, 5}, {32142, 64854, 11539}, {62188, 67924, 3}, {63414, 64100, 62104}

---

X(73098) = HATZIPOLAKIS-MOSES-MALFATTI POINT

Barycentrics    Sin[A]*(3 + Tan[B/4]^2*Tan[C/4]^2 + Tan[A/4]*(-2 + 2*Tan[C/4] + Tan[B/4]*(2 + 2*Tan[C/4]) + Tan[A/4]*(4 + Tan[B/4]*(2 - Tan[B/4] - 2*Tan[C/4]) + (2 - Tan[C/4])*Tan[C/4]))) : :

Antreas Hatzipolakis and Peter Moses, euclid 10390.

X(73098) lies on these lines:{1, 167}, {40, 483}, {164, 1489}, {1127, 1129}, {8225, 12523}

X(73098) = {X(1127),X(1129)}-harmonic conjugate of X(21465)

---

A CYCLOLOGIC THEOREM RELATED TO EXCENTRAL TRIANGLE.

[APH]

Excentral version

Let ABC be a triangle, IaIbIc the excentral triangle and P a point..

Denote

Pa, Pb, Pc = same to P points of IaBC, IbCA, IcAB, resp.

ABC, PaPbPc are circumcyclologic

Cyclologic center (ABC, PaPbPc) = Q = ? (on the circumcircle of ABC)
Cyclologic center (PaPbPc, ABC) = Q* = ? (on the circumcircle of PaPbPc)

[Ercole Suppa]

1. P on the Euler Line:

Q = X(100)
Locus of Q* as P moves on the Euler line: K086 2. P on the Brocard axis:

Q = X(101)

[Bernard Gibert]

If P is on a line through X(3) and a strong point M then the locus seems to be a circular cK(#X1,R) with singular focus F.

When M = X2, you get K086.

When M = X6, you get K040.

[APH]

For the case of the cyclologic center (ABC, PaPbPc) M can be any point strong or not.
That is:
Let ABC be a triangle, IaIbIc the excentral triangle, M a fixed point and P a point on the line OM.

Denote:

Pa, Pb,Pc = same to P points of IaBC,IbCA,IcAB, resp.

The triangles ABC, PaPbPc are circumcyclologic.

As P moves on the line OM:
The cyclologic center (ABC, PaPbPc) is a fixed point Q on the circumcircle.
The locus of the cyclologic center (PaPbPc, ABC) is a cubic.

Let's see the Q's. The class of the cubics is a subject of Bernard Gibert.
1. P on the Euler line
Q = X(100) = Reflection point of IO = X(1)X(3) line = Reflection point of Euler line of INTOUCH triangle (pedal triangle of I).

2. P on the Brocard axis
Q = X(101) = Reflection point of X(1)X(7) line = Reflection point of Brocard axis of INTOUCH triangle (pedal triangle of I).

Generalization

Let M be a fixed Point and P be a point on the line OM = L
Denote: P' = the same to P point of the INTOUCH triangle.
L' = the same to L line of the INTOUCH triangle.

Then Q is the reflection point of the L' line = IP' line of ABC

Note:
Reflection point of a line L:
The reflections La, Lb, Lc of L in the siedelines BC, CA, AB, resp. bound a triangle A*B*C*.
ABC, A*B*C* are perspective.The perspector, lying on the circumcircle, is called "Reflection pont of the line L"
It is the incenter (or an excenter) of the triangle A*B*C*.

Case of OM with M = I= X(1)

1. P = X(1) = O of INTOUCH triangle.
Pa, Pb, Pc = X(1) of IaBC, IbCA, IcAB, resp.
Cyclologic center (ABC, PaPbPc) = Q

2. P = X(354) = X(2) of the INTOUCH triangle
Pa, Pb, Pc = X(2) of the intouch triangles of IaBC, IbCA, IcAB, resp.
Cyclologic center (ABC, PaPbPc) = Q1

3. P = X(942) = X(5) of the INTOUCH triangle
Pa, Pb, Pc = X(5) of the intouch triangles of IaBC, IbCA, IcAB, resp.
Cyclologic center (ABC, PaPbPc) = Q2

Q, Q1, Q2 coincide.

Q is the reflection point of the IO line of the INTOUCH triangle.
It is the line passing thrpugh the incenters of ABC and the intouch triangle.

A CYCLOLOGIC THEOREM RELATED TO ORTHIC TRIANGLE

[APH]

Orthic Version

Let ABC be a triangle, HaHbHc the orthic triangle and P a point.

Denote:

Pa, Pb, Pc = same to P points of AHbHc, BHcHa, CHaHb, resp.

HaHbHc, PaPbPc are circumcyclologic

Cyclologic center (HaHbHc, PaPbPc = ? (on the circumcircle of HaHbHc = NPC)
Cyclologic center (PaPbPc, HaHbHc) = ? (on the circumcircle of PaPbPc)

[Ercole Suppa]

cyclologic center (HaHbHc, PaPbPc) = Poncelet point(isogonal conjugate(P))

cyclologic center (PaPbPc, HaHbHc) = orthoassociate(isogonal conjugate(circumcircleInverse(P)))

Euclid 10026

LOCUS PROBLEM

Problem by Antreas Hatzipolakis
Solution by Francisco Javier García Capitán

ETC LISTING OF Q

X(72803)

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)

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}

---

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