LOCUS PROBLEM

Problem by Antreas Hatzipolakis
Solution by Francisco Javier García Capitán

ETC LISTING OF Q

X(72803)

ETC

X(72846) = X(2)X(39)∩X(3)X(543)

Barycentrics    a^4-a^2*b^2+b^4-a^2*c^2-4*b^2*c^2+c^4 : :
X(72846) = 3*X[2]-X[34511], X[2]+X[63955], X[76]+X[13085], X[76]+2*X[32189], 4*X[3934]-X[8149], X[13085]-2*X[32189], 5*X[31276]+X[31981], X[34511]+3*X[63955], 2*X[3]-3*X[5569], X[3]-3*X[7610], 2*X[3]-X[34504], X[3]+X[34505], X[3]-2*X[34506], X[3]+3*X[40727], X[3]+2*X[63924], 5*X[3]-3*X[66616], X[5569]-2*X[7610], 3*X[5569]-X[34504], 3*X[5569]+2*X[34505], 3*X[5569]-4*X[34506], X[5569]+2*X[40727], 3*X[5569]+4*X[63924], 5*X[5569]-2*X[66616], 6*X[7610]-X[34504], 3*X[7610]+X[34505], 3*X[7610]-2*X[34506], X[7610]+X[40727], 3*X[7610]+2*X[63924], 5*X[7610]-X[66616], X[34504]+2*X[34505], X[34504]-4*X[34506], X[34504]+6*X[40727], X[34504]+4*X[63924], 5*X[34504]-6*X[66616], X[34505]+2*X[34506], X[34505]-3*X[40727], X[34505]-2*X[63924], 5*X[34505]+3*X[66616], 2*X[34506]+3*X[40727], X[34506]+X[63924], 10*X[34506]-3*X[66616], 3*X[40727]-2*X[63924], 5*X[40727]+X[66616], 10*X[63924]+3*X[66616], X[4]-3*X[7615], X[4]+2*X[7780], X[4]-2*X[47617], X[4]+3*X[63029], 2*X[4]+X[63935], 3*X[7615]+2*X[7780]

Antreas Hatzipolakis and Ercole Suppa, euclid 9964.

X(72846) lies on these lines: {2, 39}, {3, 543}, {4, 3849}, {5, 524}, {17, 9763}, {18, 9761}, {20, 7620}, {30, 5171}, {32, 8370}, {69, 625}, {83, 5346}, {99, 17004}, {115, 183}, {126, 20481}, {140, 7622}, {141, 7844}, {148, 7771}, {187, 11185}, {193, 31415}, {230, 3734}, {315, 31173}, {316, 9939}, {325, 17131}, {376, 46893}, {381, 754}, {382, 53144}, {384, 8859}, {385, 5475}, {498, 69255}, {499, 69257}, {546, 20112}, {574, 37688}, {597, 5305}, {598, 6179}, {599, 626}, {620, 37637}, {631, 1153}, {632, 12040}, {671, 1078}, {732, 38317}, {736, 7697}, {940, 50262}, {1150, 50276}, {1352, 67872}, {1506, 7754}, {1656, 7764}, {1916, 54816}, {1968, 37855}, {1975, 2482}, {1989, 9462}, {1992, 2548}, {2176, 37854}, {2549, 32457}, {2996, 60220}, {3053, 11159}, {3054, 6390}, {3090, 7758}, {3091, 7843}, {3146, 66699}, {3314, 14061}, {3363, 7745}, {3406, 60181}, {3523, 53142}, {3525, 9741}, {3526, 7619}, {3528, 53143}, {3543, 47102}, {3545, 63939}, {3552, 26613}, {3589, 42852}, {3628, 9771}, {3785, 7842}, {3793, 53418}, {3815, 7798}, {3818, 8177}, {3832, 23334}, {3839, 44678}, {3843, 63938}, {3845, 32151}, {3850, 63937}, {3851, 63927}, {3933, 7862}, {3972, 63047}, {4045, 15271}, {4197, 7621}, {4396, 7951}, {4400, 7741}, {4713, 17734}, {5007, 16924}, {5008, 63048}, {5025, 7854}, {5032, 32987}, {5054, 8716}, {5055, 9766}, {5056, 68325}, {5066, 63940}, {5068, 63929}, {5072, 63936}, {5077, 7830}, {5108, 6077}

X(72846) = midpoint of X(i) and X(j) for these {i,j}: {2, 63955}, {3, 34505}, {76, 13085}, {381, 8667}, {3543, 47102}, {5485, 7618}, {7610, 40727}, {7615, 63029}, {7620, 8182}, {7751, 7775}, {7759, 63953}, {7780, 47617}, {9740, 66466}, {9766, 63954}, {34506, 63924}, {47101, 63957}, {63952, 63956}
X(72846) = reflection of X(i) in X(j) for these {i,j}: {3, 34506}, {4, 47617}, {376, 46893}, {5569, 7610}, {7617, 16509}, {7618, 1153}, {7622, 15597}, {7759, 7775}, {7775, 5}, {8176, 7617}, {9770, 66511}, {11165, 7619}, {13085, 32189}, {34504, 3}, {34505, 63924}, {63934, 63953}, {63948, 63952}, {63952, 8667}, {63953, 7751}, {63956, 381}, {63957, 18546}, {66587, 53144}
X(72846) = complement of X(34511)
X(72846) = perspector of the circumconic through X(670)and X(9133)
X(72846) = QAP42: QA-Orthopole Center of X(43532)
X(72846) = intersection, other than A, B, C, of the circumconics: {{A,B,C,X(2),X(34154)}, {A,B,C,X(76),X(56057)}, {A,B,C,X(305),X(11167)}, {A,B,C,X(308),X(7801)}, {A,B,C,X(1989),X(7757)}, {A,B,C,X(2396),X(54843)}, {A,B,C,X(3266),X(60128)}, {A,B,C,X(3978),X(54816)}, {A,B,C,X(7607),X(11059)}, {A,B,C,X(7799),X(9462)}}
X(72846) = center of circles {{X(i),X(j),X(k)}} for these {i,j,k}: {2, 63955, 67720}, {3, 15098, 34505}, {76, 13085, 67818}, {381, 8667, 9169}, {11258, 12188, 13519}
X(72846) = pole of the line {39501, 61752} with respect to Brocard 1st circle
X(72846) = pole of the line {2793, 21006} with respect to circumcircle
X(72846) = pole of the line {32472, 47587} with respect to orthoptic circle of Steiner inellipse
X(72846) = pole of the line {2489, 8704} with respect to polar circle
X(72846) = pole of the line {1499, 38317} with respect to Warren G-circle
X(72846) = pole of the line {2793, 15561} with respect to Warren reflection circle
X(72846) = pole of the line {141, 574} with respect to Kiepert hyperbola
X(72846) = pole of the line {4576, 48974} with respect to Kiepert parabola
X(72846) = pole of the line {32, 11422} with respect to Stammler hyperbola
X(72846) = pole of the line {512, 9134} with respect to Steiner inellipse
X(72846) = pole of the line {1368, 22110} with respect to dual conic of Moses HK-parabola
X(72846) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 76, 7801}, {2, 3767, 7817}, {2, 7739, 44562}, {2, 7801, 3788}, {2, 7817, 7834}, {2, 14568, 5309}, {2, 19570, 7757}, {2, 63955, 538}, {3, 7610, 34506}, {3, 34505, 543}, {3, 34506, 5569}, {3, 40727, 34505}, {4, 7615, 47617}, {4, 7780, 63935}, {5, 7751, 7759}, {5, 7775, 8176}, {69, 43620, 625}, {76, 7746, 3788}, {76, 13085, 538}, {115, 183, 7761}, {115, 7810, 7841}, {140, 63923, 7781}, {141, 43291, 7844}, {183, 7841, 7810}, {193, 69407, 31415}, {230, 64093, 3734}, {315, 33006, 31173}, {381, 8667, 754}, {385, 33013, 7812}, {385, 69387, 5475}, {546, 63928, 63931}, {597, 8367, 7808}, {598, 6179, 34604}, {599, 11318, 626}, {599, 13881, 11318}, {626, 5461, 11318}, {671, 1078, 7833}, {671, 7833, 7748}, {1656, 63933, 7764}, {2549, 34229, 72111}, {3091, 14023, 7843}, {3767, 3934, 7834}, {3767, 32828, 3934}, {3934, 7817, 2}, {5055, 63954, 9766}, {5305, 8367, 597}, {5569, 34504, 3}, {6392, 31401, 32450}, {6392, 32838, 31401}, {7610, 34505, 3}, {7610, 40727, 543}, {7610, 63924, 34504}, {7615, 63029, 3849}, {7617, 7751, 7775}, {7617, 7775, 5}, {7620, 8182, 32479}, {7746, 7801, 2}, {7751, 7759, 63934}, {7751, 7775, 524}, {7754, 69412, 1506}, {7759, 8176, 7775}, {7759, 63953, 524}, {7780, 47617, 3849}, {7810, 7841, 7761}, {7812, 33013, 5475}, {7812, 69387, 33013}, {7828, 31276, 7822}, {7883, 9166, 5025}, {8176, 63953, 7759}, {8370, 22329, 32}, {8667, 63956, 63948}, {9740, 66466, 63942}, {11185, 17008, 187}, {11318, 13881, 5461}, {11318, 69381, 599}, {13468, 18546, 47101}, {13881, 69381, 626}, {16044, 34604, 598}, {22329, 59635, 8370}, {31173, 39565, 33006}, {32457, 72111, 2549}, {33482, 33483, 76}, {34505, 34506, 34504}, {34505, 40727, 63924}, {34506, 63924, 543}, {34508, 34509, 34507}, {37637, 69380, 620}, {37688, 47286, 574}, {47101, 63957, 30}, {47617, 63029, 63935}, {63952, 63956, 754}

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X(72847) = X(2)X(37)∩X(141)X(24209)

Barycentrics    b*c*(2*a^3-a^2*b-2*a*b^2+2*b^3-a^2*c+2*a*b*c-2*b^2*c-2*a*c^2-2*b*c^2+2*c^3) : :

Antreas Hatzipolakis and Ercole Suppa, euclid 9964.

X(72847) lies on these lines: {2, 37}, {141, 24209}, {518, 34507}, {1441, 17376}, {1733, 49484}, {2323, 4361}, {3262, 17372}, {4416, 4957}, {4641, 20886}, {4852, 68996}, {4858, 17348}, {5428, 64728}, {7232, 17885}, {7263, 16608}, {15492, 18151}, {16732, 17345}, {17235, 17861}, {17351, 20236}, {24208, 49741}, {37230, 64088}

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X(72848) = X(1)X(2)∩X(674)X(34507)

Barycentrics    a^6*b-a^5*b^2-2*a^4*b^3+2*a^3*b^4+a^2*b^5-a*b^6+a^6*c-3*a^4*b^2*c+3*a^2*b^4*c-b^6*c-a^5*c^2-3*a^4*b*c^2+2*a^3*b^2*c^2+a^2*b^3*c^2+a*b^4*c^2-b^5*c^2-2*a^4*c^3+a^2*b^2*c^3+2*b^4*c^3+2*a^3*c^4+3*a^2*b*c^4+a*b^2*c^4+2*b^3*c^4+a^2*c^5-b^2*c^5-a*c^6-b*c^6 : :

Antreas Hatzipolakis and Ercole Suppa, euclid 9964.

X(72848) lies on these lines: {1, 2}, {674, 34507}, {37823, 68366}

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X(72849) = X(25)X(2934)∩X(30)X(5171)

Barycentrics    a^10-a^6*b^4+a^4*b^6-b^10-8*a^6*b^2*c^2+5*a^4*b^4*c^2+3*b^8*c^2-a^6*c^4+5*a^4*b^2*c^4-2*b^6*c^4+a^4*c^6-2*b^4*c^6+3*b^2*c^8-c^10 : :

Antreas Hatzipolakis and Ercole Suppa, euclid 9964.

X(72849) lies on these lines: {25, 2934}, {30, 5171}, {66, 69}, {1368, 23333}, {2386, 14791}, {38736, 39860}, {39653, 44518}, {61682, 62954}

X(72849) = pole of the line {2506, 51513} with respect to orthic inconic
X(72849) = pole of the line {206, 44180} with respect to Stammler hyperbola

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X(72850) = X(25)X(2453)∩X(69)X(110)

Barycentrics    a^16-3*a^14*b^2+a^12*b^4+5*a^10*b^6-5*a^8*b^8-a^6*b^10+3*a^4*b^12-a^2*b^14-3*a^14*c^2+9*a^12*b^2*c^2-7*a^10*b^4*c^2+2*a^8*b^6*c^2+4*a^6*b^8*c^2-10*a^4*b^10*c^2+6*a^2*b^12*c^2-b^14*c^2+a^12*c^4-7*a^10*b^2*c^4-a^6*b^6*c^4+13*a^4*b^8*c^4-8*a^2*b^10*c^4+2*b^12*c^4+5*a^10*c^6+2*a^8*b^2*c^6-a^6*b^4*c^6-12*a^4*b^6*c^6+3*a^2*b^8*c^6+b^10*c^6-5*a^8*c^8+4*a^6*b^2*c^8+13*a^4*b^4*c^8+3*a^2*b^6*c^8-4*b^8*c^8-a^6*c^10-10*a^4*b^2*c^10-8*a^2*b^4*c^10+b^6*c^10+3*a^4*c^12+6*a^2*b^2*c^12+2*b^4*c^12-a^2*c^14-b^2*c^14 : :

Antreas Hatzipolakis and Ercole Suppa, euclid 9964.

X(72850) lies on these lines: {25, 2453}, {69, 110}, {1560, 52120}, {2934, 15959}, {7502, 14655}, {10201, 14676}, {14672, 18531}

X(72850) = pole of the line {110, 4235} with respect to orthoptic circle of Stammler hyperbola

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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)

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

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)

Another relationship between Napoleon cubic and Neuberg cubic

Another relationship between Napoleon cubic K005 and Neuberg cubic K001
The world of Triangle Geometry is very intrincate. There are many paths that lead to the same place.

In this case a problem from proposed by Benjamin L. Warren at Euclid 8052 and later expanded by Antreas Hatzipolakis at Euclid 8057 lead to a relationship between these two cubics.

Another relationship between Napoleon cubic and Neuberg cubic

Francisco Javier García Capitán