ETC

X(68423) = X(3)X(26913)∩X(30)X(5892)

Barycentrics    2*a^10 - 4*a^8*b^2 + a^6*b^4 + a^4*b^6 + a^2*b^8 - b^10 - 4*a^8*c^2 - 2*a^6*b^2*c^2 + 8*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 + 8*a^4*b^2*c^4 + 8*a^2*b^4*c^4 - 2*b^6*c^4 + a^4*c^6 - 5*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(68423) = X[20] + 2 X[15807], 3 X[13364] - 2 X[66531], 2 X[140] + X[13470], X[140] - 4 X[44862], X[13470] + 8 X[44862], X[143] - 4 X[64038], X[568] + 3 X[67338], 5 X[632] + X[11750], 2 X[1216] + X[11264], 11 X[3525] + X[65149], 2 X[3628] + X[44829], X[3853] + 2 X[17712], 11 X[5070] + X[64718], 2 X[5447] + X[45970], X[6146] + 2 X[32142], X[7553] - 4 X[18874], 4 X[11017] - X[16655], 2 X[11591] + X[45732], 4 X[11592] - X[68018], X[11819] - 4 X[32205], 2 X[12241] + X[63414], X[12278] - 13 X[61811], X[12289] + 11 X[15720], 2 X[12362] + X[13630], X[12897] + 2 X[44245], X[13403] + 2 X[33923], X[13419] - 4 X[35018], X[15644] + 2 X[43575], 5 X[15712] + X[21659], 4 X[16239] - X[45286], 2 X[18128] + X[31834], X[18564] + 3 X[20791], X[34798] - 7 X[66606], 13 X[46219] - X[64032], 7 X[55856] - X[61139], 5 X[61940] - 2 X[67322]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68423) lies on these lines: {2, 34513}, {3, 26913}, {5, 22352}, {20, 15807}, {30, 5892}, {140, 13470}, {143, 64038}, {265, 15246}, {389, 44056}, {539, 44324}, {547, 44407}, {549, 30522}, {550, 61744}, {568, 67338}, {632, 11750}, {1154, 11245}, {1216, 11264}, {1853, 7514}, {1899, 33533}, {3153, 13339}, {3525, 65149}, {3628, 44829}, {3819, 32423}, {3853, 17712}, {5012, 51391}, {5066, 29012}, {5070, 64718}, {5092, 46029}, {5447, 45970}, {5944, 59648}, {6146, 32142}, {6643, 32046}, {6676, 20304}, {7502, 61645}, {7553, 18874}, {8703, 32225}, {10610, 37452}, {10691, 54044}, {10984, 67869}, {11017, 16655}, {11585, 58407}, {11591, 45732}, {11592, 68018}, {11801, 55674}, {11819, 32205}, {12022, 54042}, {12100, 17702}, {12241, 63414}, {12278, 61811}, {12289, 15720}, {12362, 13630}, {12897, 44245}, {13154, 64037}, {13353, 61715}, {13391, 67336}, {13403, 33923}, {13419, 35018}, {14805, 31101}, {15311, 52073}, {15644, 43575}, {15712, 21659}, {16239, 45286}, {16252, 23060}, {18128, 31834}, {18377, 37515}, {18564, 20791}, {25739, 54006}, {34798, 66606}, {34826, 37126}, {35268, 44270}, {36201, 61574}, {37513, 37938}, {37649, 47341}, {43650, 44288}, {43821, 67321}, {44882, 46030}, {46219, 64032}, {48889, 50134}, {51392, 61659}, {55856, 61139}, {61134, 67861}, {61940, 67322}

X(68423) = midpoint of X(i) and X(j) for these {i,j}: {550, 61744}, {12022, 54042}
X(68423) = reflection of X(54044) in X(10691)

---

X(68424) = X(3)X(26913)∩X(4)X(567)

Barycentrics    2*a^10 - 4*a^8*b^2 + a^6*b^4 + a^4*b^6 + a^2*b^8 - b^10 - 4*a^8*c^2 + 6*a^6*b^2*c^2 - 5*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 + 8*a^2*b^4*c^4 - 2*b^6*c^4 + a^4*c^6 - 5*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(68424) = 3 X[4] + X[65149], 6 X[15807] + X[65149], X[12897] - 3 X[13403], 2 X[12897] + 3 X[13470], 2 X[13403] + X[13470], 3 X[13403] + X[44829], 3 X[13470] - 2 X[44829], 3 X[381] + X[12289], 5 X[1656] - X[12278], X[1885] + 2 X[11565], X[3627] - 3 X[61744], X[11750] + 3 X[61744], 3 X[3830] + X[64718], 5 X[3843] - X[64032], 3 X[3845] - X[61139], X[45732] + 2 X[52070], X[5876] - 3 X[52069], X[44076] + 3 X[52069], X[5889] + 3 X[18564], 3 X[5890] + X[18562], 3 X[5890] - X[34798], 3 X[5946] - X[6240], X[6101] - 3 X[67337], X[6102] - 3 X[12022], 3 X[12022] + X[18563], 5 X[10574] - X[18565], X[11819] - 3 X[16657], 3 X[16657] - 2 X[67867], 3 X[12100] - 4 X[44862], 3 X[13363] - 2 X[31833], 3 X[13364] - 2 X[31830], 3 X[13451] - 4 X[40240], 2 X[14128] - 3 X[34664], X[14516] - 3 X[15060], 3 X[14893] - 2 X[67322], 5 X[15026] - 3 X[38321], 7 X[15043] + X[40242], 3 X[18435] + X[34799], 4 X[18874] - 3 X[67237], X[34783] + 3 X[67339], X[34797] - 5 X[37481]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68424) lies on these lines: {3, 26913}, {4, 567}, {5, 13367}, {6, 52843}, {20, 18952}, {30, 143}, {54, 18403}, {113, 10619}, {140, 6723}, {156, 19467}, {184, 67869}, {186, 43821}, {265, 14118}, {381, 9707}, {403, 5944}, {539, 31834}, {546, 8254}, {550, 18555}, {569, 44263}, {578, 18377}, {1154, 12370}, {1199, 10296}, {1493, 66727}, {1503, 32137}, {1656, 12278}, {1658, 18390}, {1885, 11565}, {1899, 32138}, {2070, 43835}, {2072, 43394}, {2777, 18128}, {3153, 37472}, {3521, 64890}, {3575, 10095}, {3627, 11750}, {3830, 64718}, {3843, 64032}, {3845, 61139}, {3850, 45286}, {3853, 44407}, {3861, 13419}, {5133, 22804}, {5462, 45971}, {5663, 6146}, {5876, 44076}, {5889, 18564}, {5890, 18562}, {5907, 32423}, {5946, 6240}, {6101, 67337}, {6102, 12022}, {6288, 35500}, {7514, 12293}, {7526, 18396}, {7564, 18405}, {7687, 44516}, {7706, 50006}, {7728, 67879}, {8718, 18325}, {10019, 12140}, {10024, 10113}, {10212, 40685}, {10224, 11430}, {10263, 12225}, {10282, 44235}, {10540, 12254}, {10574, 18565}, {10627, 12362}, {10733, 61134}, {11264, 13754}, {11424, 44288}, {11572, 33332}, {11591, 44665}, {11819, 16657}, {12038, 49673}, {12041, 35491}, {12100, 44862}, {12106, 34785}, {12134, 45958}, {12162, 45731}, {12295, 64179}, {12429, 64105}, {12902, 34864}, {13142, 13421}, {13353, 34007}, {13363, 31833}, {13364, 31830}, {13406, 18475}, {13451, 40240}, {13491, 18560}, {13561, 18570}, {13567, 44242}, {13619, 43816}, {13861, 17845}, {14128, 34664}, {14130, 25739}, {14516, 15060}, {14644, 51033}, {14893, 67322}, {15026, 38321}, {15033, 31724}, {15043, 40242}, {15516, 29012}, {15646, 43817}, {17712, 62144}, {18364, 38724}, {18378, 41482}, {18383, 39504}, {18388, 18567}, {18435, 34799}, {18445, 66733}, {18474, 63682}, {18874, 67237}, {18912, 66721}, {18945, 32140}, {31726, 52525}, {31861, 64037}, {32142, 68018}, {32210, 67902}, {34148, 51391}, {34782, 46030}, {34783, 67339}, {34797, 37481}, {35480, 36753}, {35603, 44438}, {36966, 43844}, {37490, 66711}, {44279, 64049}, {46031, 64063}, {52008, 67067}, {61574, 61608}

X(68424) = midpoint of X(i) and X(j) for these {i,j}: {5, 21659}, {3627, 11750}, {5876, 44076}, {6102, 18563}, {6146, 52070}, {10263, 12225}, {12162, 45731}, {12370, 12605}, {12897, 44829}, {13491, 18560}, {18562, 34798}
X(68424) = reflection of X(i) in X(j) for these {i,j}: {4, 15807}, {143, 12241}, {389, 43575}, {3575, 10095}, {10627, 12362}, {11264, 45970}, {11591, 52073}, {11819, 67867}, {12134, 45958}, {13419, 3861}, {13421, 13142}, {45286, 3850}, {45732, 6146}, {45971, 5462}, {62144, 17712}, {68018, 32142}
X(68424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 13367, 58435}, {54, 18403, 67861}, {265, 14118, 34826}, {3153, 43818, 37472}, {5890, 18562, 34798}, {5944, 43865, 403}, {10113, 10610, 10024}, {11750, 61744, 3627}, {11819, 16657, 67867}, {12022, 18563, 6102}, {13403, 44829, 12897}, {18570, 67903, 13561}, {18945, 49669, 32140}, {44076, 52069, 5876}

---

X(68425) = X(3)X(26913)∩X(5)X(1495)

Barycentrics    2*a^10 - 4*a^8*b^2 + a^6*b^4 + a^4*b^6 + a^2*b^8 - b^10 - 4*a^8*c^2 + 2*a^6*b^2*c^2 + 4*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 + 4*a^4*b^2*c^4 + 8*a^2*b^4*c^4 - 2*b^6*c^4 + a^4*c^6 - 5*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(68425) = 3 X[5] + X[11750], 5 X[5] - X[61139], X[11750] - 3 X[13470], 5 X[11750] + 3 X[61139], 5 X[13470] + X[61139], 5 X[15807] + 4 X[17712], 3 X[547] - X[45286], 3 X[549] + X[21659], 3 X[12362] + X[13292], 7 X[3090] + X[65149], 7 X[3526] + X[12289], X[3575] - 3 X[13363], 7 X[3851] + X[64718], 9 X[5054] - X[12278], 9 X[5055] - X[64032], 3 X[5066] - X[13419], 3 X[5891] + X[45731], 3 X[5892] - X[45971], 3 X[5946] + X[12225], X[6101] + 3 X[12022], X[6102] + 3 X[67337], 5 X[10574] + 3 X[18564], 5 X[10574] - X[34798], 3 X[18564] + X[34798], X[11819] - 3 X[13364], 4 X[12046] - 3 X[23410], X[13491] + 3 X[52069], 3 X[15060] + X[34224], 3 X[15067] + X[44076], X[15704] + 3 X[61744], X[18562] + 7 X[66606], X[18565] - 9 X[20791], 3 X[34664] - X[45959], X[34797] - 9 X[40280], X[37484] - 9 X[67338], X[63414] - 3 X[67336]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68425) lies on these lines: {3, 26913}, {4, 37471}, {5, 1495}, {30, 5462}, {54, 51391}, {68, 33533}, {140, 30522}, {182, 18377}, {265, 37126}, {511, 43575}, {542, 45734}, {546, 44829}, {547, 45286}, {548, 13403}, {549, 21659}, {1154, 12362}, {1216, 45970}, {1503, 45958}, {2072, 10610}, {3090, 65149}, {3153, 13353}, {3292, 36966}, {3526, 12289}, {3530, 17702}, {3575, 13363}, {3581, 43816}, {3628, 18400}, {3850, 44407}, {3851, 64718}, {3856, 67322}, {3861, 29012}, {5012, 67861}, {5054, 12278}, {5055, 64032}, {5066, 13419}, {5562, 11264}, {5663, 52073}, {5876, 45732}, {5891, 45731}, {5892, 45971}, {5946, 12225}, {6101, 12022}, {6102, 67337}, {6146, 11591}, {6288, 7550}, {6756, 18874}, {7505, 34513}, {7512, 43821}, {7514, 67878}, {7516, 18396}, {7525, 18390}, {7542, 20304}, {7574, 13434}, {10116, 31834}, {10282, 50140}, {10574, 18564}, {10619, 40111}, {10627, 12370}, {11565, 14128}, {11645, 46852}, {11793, 32423}, {11801, 34004}, {11819, 13364}, {12041, 34005}, {12046, 23410}, {12103, 12897}, {12241, 13391}, {12605, 13630}, {13336, 44263}, {13339, 34007}, {13491, 52069}, {14788, 22804}, {15060, 34224}, {15067, 44076}, {15361, 43836}, {15704, 61744}, {15800, 34545}, {16266, 18536}, {18381, 49671}, {18403, 61134}, {18475, 49673}, {18531, 32046}, {18562, 66606}, {18565, 20791}, {19155, 48906}, {22352, 61750}, {25739, 34864}, {31724, 43651}, {31830, 32205}, {32142, 44665}, {32348, 36253}, {34664, 45959}, {34797, 40280}, {34826, 35921}, {37452, 43394}, {37477, 43818}, {37484, 67338}, {37514, 52843}, {63414, 67336}, {64049, 67869}

X(68425) = midpoint of X(i) and X(j) for these {i,j}: {5, 13470}, {546, 44829}, {548, 13403}, {1216, 45970}, {5562, 11264}, {5876, 45732}, {6146, 11591}, {10116, 31834}, {10627, 12370}, {11565, 14128}, {12103, 12897}, {12605, 13630}
X(68425) = reflection of X(i) in X(j) for these {i,j}: {3530, 44862}, {6756, 18874}, {12006, 64038}, {31830, 32205}, {67322, 3856}
X(68425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2072, 10610, 58407}, {10574, 18564, 34798}, {18475, 49673, 58435}

---

X(68426) = X(3)X(26913)∩X(343)X(550)

Barycentrics    2*a^10 - 4*a^8*b^2 + a^6*b^4 + a^4*b^6 + a^2*b^8 - b^10 - 4*a^8*c^2 + 14*a^6*b^2*c^2 - 8*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 - 8*a^4*b^2*c^4 + 8*a^2*b^4*c^4 - 2*b^6*c^4 + a^4*c^6 - 5*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(68426) = 5 X[13419] - 7 X[45286], 3 X[550] - X[11750], 5 X[632] - 3 X[61744], 3 X[2979] + X[18565], 3 X[3534] + X[12278], X[6101] - 3 X[54040], 5 X[6102] - 3 X[41628], 3 X[8703] - X[21659], 2 X[10095] - 3 X[66614], X[10263] - 3 X[38323], X[12289] - 5 X[15696], 2 X[12362] - 3 X[54044], 2 X[13292] - 3 X[13630], X[13292] - 3 X[31829], 3 X[13340] + X[34797], X[13491] - 3 X[44458], 3 X[14855] - X[45731], 3 X[15067] - X[18560], 3 X[15681] + X[64032], 3 X[15690] - 2 X[17712], 3 X[16657] - 4 X[32205], 3 X[16836] - 2 X[43575], 5 X[17538] - X[65149], X[18563] - 3 X[54042], 4 X[44862] - 5 X[61790], 5 X[62131] - X[64718]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68426) lies on these lines: {2, 15807}, {3, 26913}, {5, 10564}, {20, 41466}, {30, 1216}, {323, 3521}, {343, 550}, {382, 15066}, {546, 53415}, {548, 13470}, {632, 61744}, {1092, 67869}, {1885, 14128}, {2071, 34826}, {2777, 31834}, {2979, 18565}, {3530, 13403}, {3534, 12278}, {3628, 12897}, {3631, 29012}, {5663, 68018}, {5876, 52071}, {6101, 54040}, {6102, 41628}, {6288, 7464}, {7512, 12121}, {8703, 21659}, {10095, 66614}, {10113, 37452}, {10263, 38323}, {11264, 40647}, {11412, 34798}, {12084, 67878}, {12103, 18400}, {12289, 15696}, {12362, 54044}, {12605, 13416}, {13292, 13630}, {13339, 43818}, {13340, 34797}, {13491, 44458}, {13565, 44236}, {14855, 45731}, {15067, 18560}, {15681, 64032}, {15690, 17712}, {15704, 61299}, {15760, 58407}, {16196, 20304}, {16657, 32205}, {16836, 43575}, {17538, 65149}, {17845, 33532}, {18325, 43598}, {18350, 51548}, {18377, 37480}, {18563, 54042}, {18952, 61113}, {24981, 44866}, {31833, 68084}, {32142, 52070}, {32171, 63631}, {32423, 46850}, {34005, 62302}, {34007, 37477}, {37636, 64180}, {38448, 38723}, {41724, 43807}, {43574, 67861}, {44241, 63734}, {44245, 44829}, {44407, 62144}, {44440, 61753}, {44665, 45732}, {44862, 61790}, {51394, 58435}, {55631, 61543}, {61139, 62155}, {62038, 67322}, {62131, 64718}, {63441, 67926}

X(68426) = midpoint of X(i) and X(j) for these {i,j}: {5876, 52071}, {11412, 34798}, {61139, 62155}
X(68426) = reflection of X(i) in X(j) for these {i,j}: {1885, 14128}, {11264, 40647}, {12897, 3628}, {13403, 3530}, {13470, 548}, {13630, 31829}, {32137, 64035}, {44829, 44245}, {52070, 32142}, {62038, 67322}, {68084, 31833}
X(68426) = anticomplement of X(15807)
X(68426) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {343, 550, 32210}, {51394, 61750, 58435}

---

X(68427) = X(3)X(26913)∩X(5)X(156)

Barycentrics    a^10 - 2*a^8*b^2 + a^6*b^4 - a^4*b^6 + 2*a^2*b^8 - b^10 - 2*a^8*c^2 + 2*a^6*b^2*c^2 + 2*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 + 2*a^4*b^2*c^4 + 6*a^2*b^4*c^4 - 2*b^6*c^4 - a^4*c^6 - 5*a^2*b^2*c^6 - 2*b^4*c^6 + 2*a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(68427) = X[6515] + 3 X[18531], X[15068] - 3 X[16072], X[31383] - 3 X[44275]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68427) lies on these lines: {2, 265}, {3, 26913}, {4, 3521}, {5, 156}, {26, 13470}, {30, 11438}, {51, 44288}, {54, 10255}, {68, 11591}, {125, 18570}, {140, 9927}, {143, 18569}, {155, 11264}, {182, 11801}, {343, 33533}, {381, 5422}, {389, 18377}, {403, 61752}, {546, 2883}, {567, 7577}, {568, 3153}, {578, 10224}, {597, 3818}, {1147, 45970}, {1154, 6515}, {1181, 67869}, {1495, 44270}, {1503, 46030}, {1514, 3845}, {1568, 61713}, {1593, 15807}, {1656, 58407}, {1853, 31861}, {1899, 5663}, {1993, 51391}, {2072, 12022}, {3060, 7574}, {3090, 6288}, {3448, 18435}, {3518, 65149}, {3534, 15361}, {3542, 11565}, {3567, 31724}, {3580, 67337}, {5012, 10254}, {5055, 41171}, {5462, 18383}, {5876, 25738}, {5890, 18403}, {5907, 18356}, {5944, 7505}, {6102, 18404}, {6143, 45622}, {6639, 10610}, {6640, 43394}, {6643, 10627}, {6644, 18396}, {6759, 44235}, {6776, 19155}, {6816, 14128}, {7386, 54044}, {7502, 63735}, {7503, 34826}, {7507, 35603}, {7514, 14852}, {7526, 13561}, {7528, 18874}, {7530, 61299}, {7544, 22804}, {7547, 36753}, {7575, 61645}, {7592, 67861}, {7706, 18376}, {7728, 67925}, {8703, 44569}, {9306, 32423}, {9544, 14643}, {9730, 13851}, {9786, 52843}, {9818, 61702}, {10113, 12099}, {10192, 68319}, {10193, 20397}, {10263, 37444}, {10297, 11245}, {10540, 62947}, {10575, 44271}, {10938, 11561}, {10984, 61750}, {11202, 44234}, {11204, 61548}, {11250, 13403}, {11425, 31283}, {11430, 61736}, {11442, 15060}, {11487, 15077}, {11572, 63672}, {11585, 12370}, {11750, 37440}, {11818, 13364}, {12006, 18379}, {12041, 35481}, {12106, 18400}, {12121, 61128}, {12241, 13371}, {12278, 43809}, {12289, 45735}, {12359, 52073}, {12362, 63734}, {12900, 61681}, {13352, 37938}, {13363, 18420}, {13367, 60780}, {13391, 14791}, {13406, 64049}, {13421, 64048}, {13621, 64032}, {13861, 64037}, {14130, 23294}, {14216, 32137}, {14516, 50143}, {14708, 67921}, {14790, 68084}, {14864, 46849}, {14915, 58483}, {15043, 18394}, {15045, 18392}, {15053, 43836}, {15061, 35473}, {15068, 16072}, {15072, 31726}, {15133, 50008}, {16227, 44283}, {17714, 44829}, {18128, 61749}, {18324, 26958}, {18350, 34799}, {18378, 64718}, {18388, 43573}, {18559, 58789}, {18563, 26879}, {18565, 43601}, {18580, 20396}, {18583, 41729}, {19357, 58435}, {19467, 32171}, {20379, 23291}, {21243, 36253}, {21659, 37814}, {22352, 44262}, {22466, 34350}, {22660, 43588}, {23293, 38724}, {23325, 39504}, {23332, 44236}, {26937, 32210}, {31101, 37477}, {31181, 44413}, {31383, 44275}, {31830, 41362}, {32138, 52070}, {32139, 45732}, {32140, 45959}, {34477, 47296}, {34513, 68085}, {34545, 62982}, {34609, 64099}, {34664, 67926}, {34782, 44232}, {34783, 43808}, {34786, 45971}, {35493, 38728}, {39503, 57136}, {40647, 44279}, {44076, 61753}, {44665, 53415}, {44961, 51733}, {46031, 61747}, {48906, 62375}, {51548, 62974}

X(68427) = midpoint of X(6644) and X(18396)
X(68427) = reflection of X(9306) in X(50140)
X(68427) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 61701, 63839}, {4, 18952, 13630}, {4, 43816, 37481}, {5, 6146, 156}, {5, 31804, 61608}, {5, 45731, 10539}, {381, 25739, 34514}, {5012, 14644, 10254}, {9730, 13851, 44263}, {10224, 43575, 578}, {13491, 43865, 4}, {18404, 18912, 6102}, {18569, 39571, 143}, {21659, 43817, 37814}, {45970, 49673, 1147}, {52070, 67902, 32138}

---

X(68428) = X(3)X(26913)∩X(30)X(11695)

Barycentrics    2*a^10 - 4*a^8*b^2 + a^6*b^4 + a^4*b^6 + a^2*b^8 - b^10 - 4*a^8*c^2 - 6*a^6*b^2*c^2 + 12*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 + 12*a^4*b^2*c^4 + 8*a^2*b^4*c^4 - 2*b^6*c^4 + a^4*c^6 - 5*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(68428) = 3 X[140] + X[44829], 5 X[140] - X[45286], 5 X[44829] + 3 X[45286], X[143] + 3 X[67336], 3 X[548] + X[12897], 3 X[549] + X[13470], 15 X[631] + X[65149], 3 X[3917] + X[11264], X[5446] + 3 X[60749], 5 X[6101] + 3 X[41628], 3 X[7667] + X[68084], 3 X[10109] - X[67322], X[10116] + 3 X[44324], 9 X[11539] - X[61139], X[11750] + 7 X[14869], X[11819] - 9 X[64730], X[12278] - 17 X[61803], X[12289] + 15 X[15693], X[12370] + 3 X[54044], X[13403] + 3 X[34200], X[13419] - 5 X[48154], 3 X[15067] + X[45732], 15 X[15694] + X[64718], 9 X[20791] - X[34798], X[21659] + 7 X[44682], 17 X[55863] - X[64032], 3 X[61744] + 5 X[62104]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68428) lies on these lines: {3, 26913}, {30, 11695}, {140, 44829}, {143, 67336}, {265, 45308}, {548, 12897}, {549, 13470}, {550, 15807}, {631, 65149}, {3530, 30522}, {3628, 61299}, {3850, 17712}, {3917, 11264}, {5092, 10224}, {5446, 60749}, {6101, 41628}, {7667, 68084}, {10109, 67322}, {10116, 44324}, {11539, 61139}, {11592, 44665}, {11750, 14869}, {11819, 64730}, {12108, 18400}, {12278,3 X[140] + X[44829], 5 X[140] - X[45286], 5 X[44829] + 3 X[45286], X[143] + 3 X[67336], 3 X[548] + X[12897], 3 X[549] + X[13470], 15 X[631] + X[65149], 3 X[3917] + X[11264], X[5446] + 3 X[60749], 5 X[6101] + 3 X[41628], 3 X[7667] + X[68084], 3 X[10109] - X[67322], X[10116] + 3 X[44324], 9 X[11539] - X[61139], X[11750] + 7 X[14869], X[11819] - 9 X[64730], X[12278] - 17 X[61803], X[12289] + 15 X[15693], X[12370] + 3 X[54044], X[13403] + 3 X[34200], X[13419] - 5 X[48154], 3 X[15067] + X[45732], 15 X[15694] + X[64718], 9 X[20791] - X[34798], X[21659] + 7 X[44682], 17 X[55863] - X[64032], 3 X[61744] + 5 X[62104] 61803}, {12289, 15693}, {12370, 54044}, {13347, 18377}, {13348, 43575}, {13391, 64038}, {13403, 34200}, {13419, 48154}, {15067, 45732}, {15606, 32165}, {15694, 64718}, {16239, 44407}, {17702, 61792}, {20304, 34002}, {20791, 34798}, {21659, 44682}, {23060, 51425}, {29012, 35018}, {32046, 37645}, {37452, 58407}, {43821, 44832}, {51391, 61134}, {55863, 64032}, {61744, 62104}

X(68428) = midpoint of X(i) and X(j) for these {i,j}: {550, 15807}, {3850, 17712}, {13348, 43575}, {15606, 32165}

---

X(68429) = X(3)X(26913)∩X(30)X(6699)

Barycentrics    2*a^10 - 4*a^8*b^2 - a^6*b^4 + 7*a^4*b^6 - 5*a^2*b^8 + b^10 - 4*a^8*c^2 + 10*a^6*b^2*c^2 - 8*a^4*b^4*c^2 + 5*a^2*b^6*c^2 - 3*b^8*c^2 - a^6*c^4 - 8*a^4*b^2*c^4 + 2*b^6*c^4 + 7*a^4*c^6 + 5*a^2*b^2*c^6 + 2*b^4*c^6 - 5*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(68428) = 3 X[3] + X[50435], X[50435] - 3 X[63839], 5 X[6699] + X[32223], X[32223] - 5 X[44673], X[186] + 3 X[15061], X[265] + 3 X[37941], 3 X[549] - X[51394], 5 X[631] - X[22115], X[1495] - 3 X[16532], X[2071] - 5 X[38728], X[2072] - 3 X[34128], X[3292] - 7 X[14869], X[3581] + 3 X[65085], 9 X[5054] - X[50461], X[5609] - 3 X[59648], 3 X[44452] - X[51425], X[7575] + 5 X[38729], 13 X[10303] - X[63720], X[11563] - 3 X[61691], X[14157] + 7 X[15057], 3 X[15055] + X[31726], 5 X[15059] - X[18403], 11 X[15720] + X[41724], X[18571] + 2 X[20397], 2 X[20396] + X[47335], X[25739] + 3 X[37955], X[34152] - 3 X[38727], 3 X[38727] + X[63735], 3 X[35489] + X[58789], 3 X[37943] - X[51548], 3 X[38793] - X[40111], 3 X[44282] - X[51403], 3 X[46451] + X[64624]

See Antreas Hatzipolakis and Peter Moses, euclid 8425.

X(68429) lies on these lines: {3, 26913}, {5, 21663}, {30, 6699}, {125, 15646}, {140, 9729}, {143, 23336}, {156, 26937}, {185, 58435}, {186, 15061}, {265, 37941}, {343, 549}, {389, 5498}, {403, 12041}, {539, 48378}, {546, 25563}, {631, 9545}, {1154, 10257}, {1192, 31283}, {1204, 60780}, {1495, 16532}, {1503, 16531}, {1511, 62302}, {1620, 52843}, {2071, 38728}, {2072, 34128}, {2777, 46031}, {3292, 14869}, {3520, 15807}, {3523, 18952}, {3581, 65085}, {3631, 50983}, {3853, 44872}, {5054, 15066}, {5449, 43615}, {5609, 59648}, {5663, 44452}, {5946, 37118}, {6000, 44234}, {6696, 32137}, {7575, 38729}, {8254, 15012}, {10018, 13491}, {10096, 14915}, {10113, 44246}, {10125, 40647}, {10151, 34584}, {10212, 43575}, {10255, 34798}, {10264, 51393}, {10303, 63720}, {10627, 16196}, {11264, 12038}, {11438, 61736}, {11563, 61691}, {11704, 18565}, {12006, 37649}, {12106, 23329}, {12108, 20585}, {13358, 45780}, {13363, 52262}, {13364, 44236}, {13391, 15122}, {13406, 43604}, {13470, 15331}, {13561, 18474}, {14157, 15057}, {15055, 31726}, {15059, 18403}, {15088, 23323}, {15311, 68319}, {15720, 41724}, {16111, 44283}, {16238, 45959}, {17702, 37968}, {17704, 34004}, {18282, 46850}, {18388, 34331}, {18400, 18571}, {20396, 47335}, {22467, 34826}, {23328, 46030}, {25739, 37955}, {26879, 43394}, {32110, 37938}, {32140, 58378}, {32171, 45732}, {32767, 45971}, {34152, 38727}, {34797, 45622}, {35489, 58789}, {35491, 43865}, {37943, 51548}, {38793, 40111}, {43584, 48411}, {43608, 45735}, {44235, 64027}, {44282, 51403}, {44911, 61574}, {46451, 64624}, {47486, 64757}, {49116, 51733}, {50143, 64180}

X(68429) = midpoint of X(i) and X(j) for these {i,j}: {3, 63839}, {5, 21663}, {125, 15646}, {403, 12041}, {6699, 44673}, {10113, 44246}, {10264, 51393}, {16111, 44283}, {32110, 37938}, {34152, 63735}, {44234, 61548}, {49116, 51733}
X(68429) = reflection of X(i) in X(j) for these {i,j}: {3853, 44872}, {23323, 15088}, {61574, 44911}
X(68429) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {140, 13630, 58407}, {140, 44158, 11591}, {1204, 60780, 67869}, {6696, 44232, 32137}, {32171, 67902, 45732}, {38727, 63735, 34152}

---

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

ΠΑΝΑΓΙΩΤΗΣ ΛΑΔΟΠΟΥΛΟΣ (1914 - 2005)

ΠΑΝΑΓΙΩΤΗΣ ΛΑΔΟΠΟΥΛΟΣ (1914 - 2005) Καθηγητής Πολυτεχνείου

ΒΙΟΓΡΑΦΙΚΑ
ΒΙΟΓΡΑΦΙΚΟ ΣΗΜΕΙΩΜΑ ΤΟΥ Π. Δ. ΛΑΔΟΠΟΥΛΟΥ

Βιογραφικές πληροφορίες

1. Συμμετοχή σε κινητοποίηση σπουδαστών του ΕΜΠ το 1934
"Ανάμεσα στους αριστερούς σπουδαστές που παραπέμφθηκαν στη Σύγκλητο, ήταν και ο Π. Λαδόπουλος, μετέπειτα καθηγητής του ΕΜΠ, ο οποίος, όταν ρωτήθηκε τι δουλειά είχε αυτός σε κομμουνιστικές συγκεντρώσεις, απάντησε ότι μετείχε όχι μόνο γιατί τον έθιγε το Διάταγμα αλλά και γιατί δεν ήθελε να αφήσει τη δόξα της αντίστασης μόνο στους κομμουνιστές."
ΠΗΓΗ: 130 ΧΡΟΝΙΑ ΣΧΟΛΗ ΠΟΛΙΤΙΚΩΝ ΜΗΧΑΝΙΚΩΝ ΕΜΠ ΜΗΤΕΡΑ–ΤΡΟΦΟΣ ΤΗΣ ΕΛΛΗΝΙΚΗΣ ΤΕΧΝΟΛΟΓΙΑΣ Ιστορική αναδρομή στην ανάπτυξη των υποδομών της Ελλάδας 1887-2017, σ. 34

2. Μίμης Ανδρουλάκης στα γεγονότα του Πολυτεχνείου. Ερωτηση στον Π. Λαδόπουλο
"Στα σκαλοπάτια έχω μια μικρή σκηνή με τον αντιπρύτανη:
«Κύριε Λαδόπουλε, αυτός είναι ο “ένας” του λεωφορείου;»
και δείχνω τον διευθυντή της Αστυνομίας.
«Τι θράσος είναι αυτό», διαμαρτύρεται έξαλλος ο Δασκαλόπουλος στον αντιπρύτανη.
Τι εννοούσα; Ο Παναγιώτης Λαδόπουλος μας έκανε παλαιότερα Προβολική Γεωμετρία και του άρεσε να επαναλαμβάνει το σόφισμα: «Αν σ’ ένα λεωφορείο οι 29 θέλουν ανοιχτό παράθυρο κι ο ένας κρυώνει και ζητά να το κλείσουν, τίνος το αίτημα είναι υπέρτερο; Της πλειοψηφίας ή του ενός;»
Τον έκοψα, κάτι σπάνιο για μένα.
«Κύριε καθηγητά, αρνείστε τον πλειοψηφικό κανόνα τηςΔημοκρατίας ή απλώς υποστηρίζετε τα εύλογα δικαιώματα της μειοψηφίας;»
Θύμωσε τότε. Ήταν το σόφισμά του μια κομψή απολογίατης δικτατορίας.
ΠΗΓΗ: ΜΙΜΗΣ ΑΝΔΡΟΥΛΑΚΗΣ, ΠΡΙΝ ΣΒΗΣΟΥΝ ΤΑ ΦΩΤΑ. Αυτοβιογραφία - Ιστορία

3. Ποινή για φιλοχουντικη στάση
46. Λαδόπουλος Π. (ΕΜΠ) 2 μήνες αργία
ΠΗΓΗ: Δημήτρη Νταβέα, Τα στοιχεία για τη συνεργασία των καθηγητών των Α.Ε.Ι. με τη δικτατορία. Ο Πολίτης, τεύχος 2, Ιούνιος 1976, σ. 24

ΒΙΒΛΙΟΓΡΑΦΙΚΑ

1933
Π. Δ. ΛΑΔΟΠΟΥΛΟΥ, ΜΕΘΟΔΙΚΑΙ ΑΣΚΗΣΕΙΣ ΟΛΟΚΛΗΡΩΤΙΚΟΥ ΛΟΓΙΣΜΟΥ. 1933 (174 σελιδες)
ΠΡΟΛΟΓΟΣ

1947
Παναγ. Δημ. Λαδοπούλου, Συμβολή εις την Προβολικήν Γεωμετρίαν. Περί Κινητότητος Γεωμετρικών Σχηματισμών. Εναίσιμος εππι διδακτοριά διατριβή... Αθήναι 1947. (78 σελίδες)

1948
Π. Δ. Λαδόπουλος, Συμβολή εις την Προβολικήν Γεωμετρίαν. Περί Κινητότητος Γεωμετρικών Σχηματισμών. Δελτίον της Ελληνική Μαθηματικής Εταιρείας 23, 51-126 (1948)
zbMATH

Παναγ. Λαδοπούλου, GEOMETRICA CEOMETRICE (Gauss). Παράρτημα του Δελτίου της Εληνικής Μαθηματικής Εταιρείας, Νοέμβριος - Δεκέμβριος 1948, σελ. 580
Σχόλιο: Το ". . . arithmetica arithmetice, geometrica geometrice doceantur" δεν είναι του Gauss, αλλά του Ramus (Βλέπε Ramus et l'Université, σελ. 76, σημ. 28).
Η φράση υπάρχει ως motto και στη μελέτη Michael Detlefsen - Andrew Arana, Purity of Methods. Philosopher's Imprint, vol. 11, no 2, January 2011, p. 1-20

Takis Ladopoulos, Some Theorems on Cyclic Polygons Inscribed in a Circle.
The American Mathematical Monthly, Vol. 55, No. 5 (May, 1948), pp. 301-307
A Clarification: Some Theorems on Cyclic Polygons Inscribed in a Circle
The American Mathematical Monthly, Vol. 56, No. 4 (Apr., 1949), p. 247

1949
Π. Δ. Λαδόπουλος, Συμβολή εις την Προβολικήν Γεωμετρίαν. Η Γωνία δύο Κωνικών Τομών. Δελτίον της Ελληνική Μαθηματικής Εταιρείας 24, 76-84 (1949)
zbMATH

1951
Ladopoulos, Panaïotis, Sur la métrique des courbes algébriques.
C. R. Acad. Sci., Paris 233, 1417-1418 (1951).
zbMATH

1952
Ladopoulos, P. D., Sur la métrique des courbes algébriques.
Bulletin de l'Académie Royale de Belgique, V. Sér. 38, 469-477 (1952).
zbMATH

1953
P. Ladopoulos, Sur la metrique des coniques. Publications scientifiques de l' Universite Nationale Technique 9. Athenes 1953

1954
Π. Δ. Λαδοπούλου, Σημειώσεις Προβολικής Γεωμετρίας. Εκδόσεις A. Καραβία, Αθήναι 1954

Ladopoulos, Panaiotis D., Une extension d’un théorème de Clifford.
R. Acad. Sci., Paris 238, 2050-2051 (1954).
zbMATH

1957
Ladopoulos, Panaiotis D., Généralisation du théorème de Desargues-Sturm.
Soc. R. Sci. Liège 26, 16-18 (1957).
zbMATH

1960
Π. Δ. Λαδοπούλου, Σημειώσεις Προβολικής Γεωμετρίας. Εκδόσεις A. Καραβία, Αθήναι 1960

Ladopoulos, Panaïotis, La géométrie euclidienne des éléments d’un réseau de coniques.
C. R. Acad. Sci., Paris 250, 811-812 (1960).
zbMATH

1962
Ladopoulos, P., Sur les courbes d’Apollonius.
Soc. R. Sci. Liège 31, 492-496 (1962).
zbMATH

1964
Π. Δ. Λαδοπούλου, Μαθήματα Προβολικής Γεωμετρίας. Εκδόσεις Α. Καραβία, Αθήναι 1964

Παν. Δ. Λαδοπούλου, Μαθήματα Παραστατικής Γεωμετρίας. Τεύχος Πρώτον. Εκδόσεις Α. Καραβία, Αθήναι 1964

1966
Παν. Δ. Λαδοπούλου, Στοιχεία Προβολικής Γεωμετρίας. Τόμος Πρώτος. Εκδόσεις Α. Καραβία, Αθήναι 1966 (σελ. ζ + 1 λ. + 210)

1968
Παν. Δ. Λαδοπούλου, Εικαστικαί Τέχναι και Γεωμετρία. Πρακτικά της Ακαδημίας Αθηνών 43(1968) Α 19 - 63.
ΨΗΦ. ΠΑΑ 1968 και ΠΑΑ 1968
ΨΗΦ. ΛΑΔΟΠΟΥΛΟΣ

Π. Δ. Λαδοπούλου, Στοιχεία Παραστατικής Γεωμετρίας Ε - ΣΤ Γυμνασίου. Ο.Ε.Σ.Β., Αθήναι 1968

1970
P. D. Ladopoulos, Fine Arts and Geometry
The Journal of Aesthetics and Art Criticism, Volume 28, Issue 4, Summer 1970, pp. 535–540

1971
Παν. Δ. Λαδοπούλου, Στοιχεία Προβολικής Γεωμετρίας. Τόμος Πρώτος. Αθήναι 1971 (σελ. ζ + 1 λ. + 209 + 1 λ.)

1972
Παν. Δ. Λαδοπούλου, Στοιχεία Προβολικής Γεωμετρίας, Τόμος Δεύτερος, τεύχος Α'. Εκδοσεις Α. Καραβία. Αθήναι 1972 (σελ. 110 + 2 λ.)
Παν. Δ. Λαδοπούλου, Στοιχεία Προβολικής Γεωμετρίας, Τόμος Δεύτερος, τεύχος Β'. (σελ. δ' + [111] - 315 + 1)

Παν. Δ. Λαδοπούλου, Μαθήματα Παραστατικής Γεωμετρίας. Τεύχος Πρώτον. Αθήναι [1972] (σελ. 200)
Παν. Δ. Λαδοπούλου, Μαθήματα Παραστατικής Γεωμετρίας Μετά Στοιχείων Διαφορικής Γεωμετρίας. Τεύχος Δεύτερον. Αθήναι [1972] (σελ. 40)

1976
Παν. Δ. Λαδοπούλου, Στοιχεία Παραστατικής Γεωμετρίας. Αθήναι 1976 (σελ. 237 + 30 έγχρωμες)

1983
P. Ladopoulos, La Géométrie Euclidienne des Eléments d’ un reseau de courbes algébriques.
?? 1983

1999
Ladopoulos, Panaiotis, Sur la mobilitè des polyèdres.
Atti Accad. Pontaniana (N.S.) 48, 345-354 (1999).
zbMATH

INVERSE IN THE CIRCUMCIRCLE

Let ABC be a triangle, O the circumcenter, P a point different to O and Q a point not lying on the OP line.
The radical axis of the circumcircles of ABC and OQP intersect the line OP at P'. P' is the inverse of P wrt circumcircle of ABC
APH, Comment in Romantics of Geometry problem

ARCHIVES

Old yahoo group Anopolis Anopolis

ENNEADIC LINES

Definition
Let T1, T2, T3 be three triangles with the property: The Nine Point Circle centers N1, N2, N3 of T1, T2, T3, resp. are collinear.
I call the line N1N2N3 enneadic line of T1, T2, T3
See an example of an enneadic line in geometry of triangle Euclid 6930

Conjecture
Let A1A2A3... An be a regular n-gon, n>4.  
There are three triangles of the n-gon having an enneadic line.

Problem
How many "different" enneadic lines are in a n-gon ?
(two enneadic lines are "different" if the 2 triads of the triangles are not symmetric)

Similarly we may ask for enneadic lines of 4 triangles.

PICTURES

Pentagon

Hexagon

Heptagon 1

Heptagon 2

Heptakaidecagon

Heptagon (4 triangles. Elias Hagos)

Mail Antreas P. Hatzipolakis

MORLEY RELATED TRIANGLE CENTERS

MORLEY RELATED TRIANGLE CENTERS
BY ANTREAS HATZIPOLAKIS and CHRIS VAN TIENHOVEN, PETER MOSES, CESAR LOZADA

*****************************************************************************

Antreas Hatzipolakis and Chris van Tienhoven

X(6120) =  1st HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6121) =  2nd HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6122) =  3rd HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6123) =  4th HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6124) =  5th HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6125) =  6th HATZIPOLAKIS-VAN TIENHOVEN POINT
X(66087) = HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE CENTER
X(66089) = PERSPECTOR OF THESE TRIANGLES: 1st MORLEY AND HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE

Antreas Hatzipolakis and Peter Moses

X(65155) =  X(2)X(15857)∩X(357)X(8065)
X(65156) =  HATZIPOLAKIS-MOSES EQUILATERAL TRIANGLE CENTER
X(65157) =  X(2)X(3276)∩X(5)X(3280)
X(65158) =  X(357)X(5456)∩X(358)X(1136)
X(65377) = CENTER OF HATZIPOLAKIS-MOSES-MORLEY HYPERBOLA
X(65378) = PERSPECTOR OF HATZIPOLAKIS-MOSES-MORLEY HYPERBOLA
X(65379) = X(357)X(1136)∩X(10632)X(16871)
X(65380) = ANTICOMPLEMENT OF X(65377)
X(65381) = 98TH HATZIPOLAKIS-MOSES-EULER POINT
X(65382) = TRILINEAR POLE OF LINE {6, 65378}
X(66329) = 1st HATZIPOLAKIS-MOSES TRISECTOR TRIANGLES PERSPECTOR
X(66330) = 2nd HATZIPOLAKIS-MOSES TRISECTOR TRIANGLES PERSPECTOR
X(66331) = 3rd HATZIPOLAKIS-MOSES TRISECTOR TRIANGLES PERSPECTOR
X(66332) = CENTER OF THE 1st MORLEY CONIC
X(66333) = PERSPECTOR OF THE 1st MORLEY CONIC
X(66365) = X(356)-CEVA CONJUGATE OF X(357)
X(66366) = X(3276)-CEVA CONJUGATE OF X(1136)
X(66367) = X(3277)-CEVA CONJUGATE OF X(1134)
X(66619) = X(3275)X(3602)∩X(3276)X(3606)

Antreas Hatzipolakis and César Lozada

X(66478) = CENTER OF THE 1st EULER-ROUSSEL EQUILATERAL TRIANGLE
X(66479) = CENTER OF THE 2nd EULER-ROUSSEL EQUILATERAL TRIANGLE
X(66480) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EULER-ROUSSEL AND 1st MORLEY
X(66481) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EULER-ROUSSEL AND 1st MORLEY

César Lozada

X(66180) = CENTER OF THE 2nd HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE
X(66181) = CENTER OF THE 3rd HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE
X(66182) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd MORLEY AND 2nd HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE
X(66183) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd MORLEY AND 3rd HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE