[Antreas P. Hatzipolakis]:
Let ABC be a triangle, A1B1C1 the pedal triangle of I, P a point and A'B'C' the pedal triangle of P.
Denote:
.
R1, R2, R3 = the reflections of PA', PB', PC' in B1C1, C1A1, A1B1, resp.
A*B*C* = the triangle bounded by R1, R2, R3
ABC, A*B*C* are parallelogic.
Parallelogic center (ABC, A*B*C*) = O = X(3)
Parallelogic center (A*B*C*, ABC) ?
Which is the locus of the parallelogic center (A*B*C*, ABC) as P moves on a line, the Euler line for example?
[Peter Moses]:
Hi Antreas,
Which is the locus of the parallelogic center (A*B*C*, ABC) as P moves on a line, the Euler line for example?
[Peter Moses]:
Hi Antreas,
>Parallelogic center (A*B*C*, ABC) ?For P = {p,q,r}
(a - b - c) ((a b - b^2 + a c + 2 b c - c^2) p + (b - c) (a - b + c) q - (b - c) (a + b - c) r) : :
Some examples:
{1,10}
{3,10916}
{4,946}
{7,142}
{8,21627}
{10,3813}
{11,11}
{30,758}
{55,4847}
{56,1210}
{57,11019}
{65,1}
{79,11263}
{80,21630}
{84,6245}
{85,20257}
{104,10265}
-----------------------------------------------------------
P = X(2);
= COMPLEMENT OF X(3158)
(a-b-c) (a b-3 b^2+a c+6 b c-3 c^2) : :
=X[3189] - 7 X[3624], 5 X[3617] + X[3680], X[10] + 2 X[3813], 4 X[3634] - X[3913], X[6601] + 2 X[6666], 5 X[3091] + X[6762], 11 X[5056] + X[6764], 7 X[3090] - X[6765], X[2136] - 7 X[9780], 2 X[3626] + X[10912], X[946] + 2 X[10916], 4 X[1125] - X[12437], 5 X[3616] + X[12625], 5 X[5818] + X[12629], 4 X[10] - X[12640], 8 X[3813] + X[12640], X[12632] - 13 X[19877], X[12513] + 2 X[19925], 4 X[3813] - X[21627], 2 X[10] + X[21627], X[12640] + 2 X[21627].
= lies on these lines : {1,6856},{2,3158},{8,18220},{9,5274},{10,496},{11,210},{142,2886},{200,10589},{226,3873},{312,17774},{321,4939},{497,4512},{516,11235},{518,3817},{519,5055},{522,21204},{527,1699},{528,10164},{758,946},{908,4661},{950,10527},{1058,5705},{1125,12437},{1210,3753},{1329,4711},{1420,5175},{1706,5704},{2136,9780},{2321,3705},{2550,6692},{2887,17059},{2900,4666},{3090,6765},{3091,6762},{3189,3624},{3243,5226},{3434,3911},{3616,12625},{3617,3680},{3626,10912},{3634,3913},{3687,4923},{3717,4903},{3755,24239},{3756,21949},{3794,17197},{3825,3956},{3838,5542},{3847,4662},{3877,5837},{3892,21620},{3894,12047},{3921,4187},{3928,9812},{3939,17123},{4031,20292},{4035,10453},{4134,21616},{4314,4999},{4525,11813},{4669,5854},{4731,8582},{4859,21267},{4863,6745},{4906,17070},{5056,6764},{5087,21060},{5325,11238},{5573,17067},{5744,9580},{5784,17626},{5795,9581},{5818,12629},{6557,10005},{6600,8167},{6601,6666},{6736,17606},{6737,11376},{7681,9842},{7741,21075},{8666,18519},{9669,12572},{10106,10529},{10896,12527},{12513,19925},{12632,19877},{17889,24216},{18483,18540}.
= complement of X(3158).
= midpoint of X(3928) and X(9812).
= reflection of X(3817) in X(3829).
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 3813, 21627), (10, 21627, 12640), (11, 4847, 3452), (497, 5231, 5745), (2886, 11019, 142), (3756, 21949, 24175), (6734, 12053, 5837).
= complement of the isogonal of X(19604).
= X(i)-complementary conjugate of X(j) for these (i,j): {7, 2885}, {56, 3161}, {269, 12640}, {1293, 4521}, {3445, 9}, {3676, 5510}, {4373, 1329}, {8056, 3452}, {10029, 20540}, {16079, 8}, {16945, 37}, {19604, 10}.
-----------------------------------------------------------
= complement of X(3158).
= midpoint of X(3928) and X(9812).
= reflection of X(3817) in X(3829).
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 3813, 21627), (10, 21627, 12640), (11, 4847, 3452), (497, 5231, 5745), (2886, 11019, 142), (3756, 21949, 24175), (6734, 12053, 5837).
= complement of the isogonal of X(19604).
= X(i)-complementary conjugate of X(j) for these (i,j): {7, 2885}, {56, 3161}, {269, 12640}, {1293, 4521}, {3445, 9}, {3676, 5510}, {4373, 1329}, {8056, 3452}, {10029, 20540}, {16079, 8}, {16945, 37}, {19604, 10}.
-----------------------------------------------------------
P = X(5);
= COMPLEMENT OF X(8715)
a^2 b^2-b^4-2 a^2 b c+a b^2 c+a^2 c^2+a b c^2+2 b^2 c^2-c^4 : :
= X[5] - 3 X[3829], X[3813] + 3 X[3829], 5 X[1656] - X[3913], 7 X[3526] - 3 X[4421], X[6765] - 9 X[7988], X[3811] - 5 X[8227], 3 X[5790] + X[10912], 3 X[10175] - X[10915], X[382] + 3 X[11194], X[3] + 3 X[11235], 7 X[3851] - 3 X[11236], 3 X[381] + X[12513], 3 X[5] - X[12607], 9 X[3829] - X[12607], 3 X[3813] + X[12607], 7 X[9624] + X[12625], 7 X[7989] + X[12629], 17 X[7486] - X[12632], X[12635] - 5 X[18493], 3 X[3817] - X[21077], 3 X[10175] + X[21627], 3 X[5886] - X[22836].
= lies on these lines : {1,2476},{2,3746},{3,11235},{4,535},{5,519},{8,3814},{10,11},{12,3244},{21,4857},{35,149},{72,11813},{78,23708},{80,4861},{115,17448},{116,20257},{140,528},{145,7951},{214,5533},{226,3881},{354,11263},{355,22837},{377,10072},{381,12513},{382,11194},{404,3582},{405,11238},{442,551},{474,10199},{495,3635},{496,1125},{497,5248},{499,3434},{501,19642},{515,10943},{516,6705},{518,9955},{529,546},{536,7764},{758,946},{956,10896},{958,9669},{960,7743},{993,1479},{1058,10198},{1210,3754},{1212,21090},{1329,3626},{1385,1484},{1478,10529},{1506,20691},{1656,3913},{1770,4973},{2475,5563},{2550,10200},{2801,12608},{2893,24202},{2975,3583},{3035,20107},{3058,7483},{3086,6904},{3120,3953},{3241,5141},{3303,10197},{3304,17532},{3337,20292},{3419,11376},{3452,4015},{3526,4421},{3555,17605},{3584,7504},{3625,7173},{3632,11681},{3634,3816},{3678,4847},{3679,4193},{3680,6975},{3811,8227},{3817,18908},{3820,3847},{3826,19878},{3828,9710},{3838,5045},{3851,11236},{3868,18393},{3872,10826},{3874,12047},{3878,6734},{3880,9956},{3889,10129},{3892,13407},{3925,19862},{3968,8582},{4085,20108},{4297,15908},{4301,6831},{4309,6910},{4324,5303},{4330,17549},{4426,9665},{4479,7796},{4658,14009},{4669,17533},{4745,9711},{4972,19864},{4999,15171},{5046,5258},{5057,6763},{5080,5288},{5082,10589},{5129,5274},{5223,7678},{5231,9614},{5267,6284},{5270,17577},{5299,17737},{5450,10525},{5497,24161},{5537,6972},{5603,6873},{5690,13463},{5777,22835},{5790,10912},{5794,11373},{5881,6941},{5882,6842},{5886,22836},{6154,7294},{6690,15172},{6765,7988},{6828,11522},{6829,9624},{6830,7982},{6871,11240},{6881,12437},{6882,11362},{6933,10056},{6943,7991},{6990,11523},{7486,12632},{7681,19925},{7752,17144},{7956,12571},{7989,12629},{9670,16370},{10175,10915},{10948,17647},{11019,12446},{11260,18480},{11928,22758},{12635,18493},{14872,21635},{14923,18395},{15888,17530},{16174,20117},{16829,17669},{17046,17761},{17529,19883},{21073,24036},{21384,24045}.
= complement of X(8715).
= midpoint of X(i) and X(j) for these {i,j}: {{4, 8666}, {5, 3813}, {355, 22837}, {946, 10916}, {5450, 10525}, {5690, 13463}, {10915, 21627}, {11260, 18480}}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 7741, 3814), (10, 11, 3825), (10, 12053, 3884), (10, 21630, 3057), (21, 10707, 4857), (496, 2886, 1125), (1125, 2886, 3841), (1479, 10527, 993), (3813, 3829, 5), (4847, 21616, 3678), (5231, 9614, 12514), (9710, 17527, 3828), (10175, 21627, 10915), (10914, 17606, 10).
-----------------------------------------------------------
= complement of X(8715).
= midpoint of X(i) and X(j) for these {i,j}: {{4, 8666}, {5, 3813}, {355, 22837}, {946, 10916}, {5450, 10525}, {5690, 13463}, {10915, 21627}, {11260, 18480}}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 7741, 3814), (10, 11, 3825), (10, 12053, 3884), (10, 21630, 3057), (21, 10707, 4857), (496, 2886, 1125), (1125, 2886, 3841), (1479, 10527, 993), (3813, 3829, 5), (4847, 21616, 3678), (5231, 9614, 12514), (9710, 17527, 3828), (10175, 21627, 10915), (10914, 17606, 10).
-----------------------------------------------------------
P = X(6);
= X(1)X(475)∩X(10)X(1001)
(a-b-c) (a^3 b-a^2 b^2+a b^3-b^4+a^3 c+2 a^2 b c-a b^2 c+2 b^3 c-a^2 c^2-a b c^2-2 b^2 c^2+a c^3+2 b c^3-c^4) : :
= lies on these lines: {1,475},{10,1001},{141,15733},{142,17059},{281,497},{515,1597},{518,12618},{522,3663},{527,21629},{740,10916},{946,1871},{1210,3755},{2321,3693},{2345,6601},{3174,17284},{3886,6734},{3939,17353},{3946,11019},{4523,21616},{4858,23529},{4953,17246},{5572,16608},{5856,17351},{10915,17765},{18216,18634}.
= X(i)-complementary conjugate of X(j) for these (i,j): {{56, 5452}, {3433, 9}, {13577, 1329}}.
-----------------------------------------------------------
= X(i)-complementary conjugate of X(j) for these (i,j): {{56, 5452}, {3433, 9}, {13577, 1329}}.
-----------------------------------------------------------
X(9);
= COMPLEMENT OF X(3174)
(a-b-c) (a^3 b-3 a^2 b^2+3 a b^3-b^4+a^3 c+2 a^2 b c-3 a b^2 c+4 b^3 c-3 a^2 c^2-3 a b c^2-6 b^2 c^2+3 a c^3+4 b c^3-c^4) : :
= X[7674] - 5 X[18230].
= lies on these lines: {2,3174},{9,497},{10,1001},{11,3059},{142,2886},{226,15185},{390,6734},{480,4863},{516,1158},{518,946},{527,11235},{528,10265},{908,7678},{960,4342},{1210,2550},{1445,3434},{3243,3485},{3826,9843},{4134,18254},{4326,5231},{4662,18255},{4686,4953},{5223,9614},{5249,11025},{5542,11263},{5745,8730},{6067,14100},{7674,18230},{7675,10527},{11680,21617},{12609,20116},{12731,21631},{17059,21255}.
= complement of X(3174).
= midpoint of X(9) and X(6601).
= reflection of X(6600) in X(6666).
= {X(2886),X(5572)}-harmonic conjugate of X(142).
= X(4578)-Ceva conjugate of X(522).
= crosssum of X(56) and X(21059).
= barycentric product X(8)X(24181).
= barycentric quotient X(24181)/X(7).
= complement of X(3174).
= midpoint of X(9) and X(6601).
= reflection of X(6600) in X(6666).
= {X(2886),X(5572)}-harmonic conjugate of X(142).
= X(4578)-Ceva conjugate of X(522).
= crosssum of X(56) and X(21059).
= barycentric product X(8)X(24181).
= barycentric quotient X(24181)/X(7).
-----------------------------------------------------------
P = X(12);
= COMPLEMENT OF X(3871)
a^2 b^2-b^4-2 a^2 b c+2 a b^2 c+a^2 c^2+2 a b c^2+2 b^2 c^2-c^4 : :
= 3 X[3584] - 4 X[6668], 2 X[12] - 3 X[17530], 3 X[17577] - X[20060], 3 X[17549] - X[20066].
= lies on these lines: {1,442},{2,496},{3,3434},{4,956},{5,8},{9,9614},{10,11},{12,519},{21,149},{30,2975},{35,528},{39,21956},{40,5231},{55,7483},{56,11112},{63,3650},{65,10916},{72,946},{75,17181},{78,5886},{92,15763},{100,140},{145,495},{200,8227},{210,21616},{226,3555},{325,17143},{354,12609},{355,1532},{377,999},{381,3436},{388,17532},{390,6857},{404,15325},{405,497},{443,14986},{474,2550},{498,3913},{499,1376},{515,15908},{516,3916},{517,6734},{518,12047},{529,3585},{546,5080},{548,5303},{551,3841},{594,17444},{631,17784},{726,21927},{867,17164},{908,9955},{942,20612},{944,5175},{952,4861},{954,6601},{958,1479},{962,5789},{993,6284},{997,11376},{1001,11517},{1056,5177},{1086,3953},{1125,3925},{1191,1714},{1210,3753},{1212,21073},{1260,6832},{1278,7906},{1319,10949},{1329,3679},{1385,10609},{1478,12513},{1519,5777},{1537,5887},{1538,9947},{1565,20880},{1621,6675},{1656,5552},{1697,5705},{1698,3816},{1737,5836},{1738,17055},{1883,4968},{2170,21029},{2475,18990},{2478,9669},{2551,10591},{2646,10959},{2932,6940},{2968,23528},{3006,3695},{3035,5533},{3058,5248},{3090,7080},{3091,3421},{3136,17135},{3244,3822},{3303,10198},{3338,5880},{3340,15844},{3452,3697},{3582,6691},{3583,5258},{3584,6668},{3614,3625},{3616,8728},{3617,3820},{3621,5141},{3622,4197},{3624,3826},{3626,3814},{3632,7951},{3649,3874},{3656,11682},{3678,11813},{3746,6690},{3811,4863},{3817,21075},{3824,5049},{3838,13407},{3847,9711},{3869,6841},{3870,11374},{3873,6147},{3878,21677},{3880,10039},{3881,11263},{3893,10915},{3927,11415},{3933,4441},{3976,17889},{4002,8582},{4026,19863},{4294,16370},{4299,11194},{4353,21955},{4413,10200},{4423,17590},{4511,5178},{4662,5087},{4678,5154},{4853,5587},{4857,5251},{4865,17733},{4875,5179},{4882,7988},{4904,17046},{4915,7989},{5044,7743},{5045,5249},{5084,5274},{5100,7081},{5173,14054},{5176,18357},{5180,11684},{5219,6765},{5226,6764},{5250,5791},{5254,16975},{5260,10707},{5267,15338},{5291,7745},{5292,5710},{5305,17737},{5432,8715},{5439,11019},{5450,11826},{5484,17677},{5499,6224},{5559,12653},{5657,6922},{5686,7678},{5690,6882},{5697,13463},{5711,11269},{5722,19860},{5724,15955},{5726,11519},{5744,6361},{5745,10624},{5815,9779},{5833,10384},{5842,11012},{5853,13411},{5854,8068},{5855,11009},{5881,18242},{5904,18393},{6174,7294},{6244,6890},{6653,7824},{6705,17613},{6735,9956},{6736,10175},{6737,13464},{6743,7958},{6762,9612},{6763,17768},{6829,10595},{6830,12245},{6833,10306},{6862,10679},{6871,9654},{6872,9668},{6889,10806},{6910,20075},{6913,10531},{6917,10680},{6929,10522},{6933,10528},{6937,7967},{6980,10942},{6984,10597},{6991,20007},{7288,16371},{7354,8666},{7373,11240},{7680,7982},{7767,20553},{8256,18395},{9578,12629},{9581,9623},{9655,20076},{9780,17527},{9874,18228},{10058,13271},{10523,10912},{10525,22758},{10572,12690},{10585,11239},{10944,22837},{11256,12749},{11373,19861},{11544,17483},{11698,12531},{12514,12701},{12527,18483},{12608,13257},{12670,21631},{14019,16020},{14740,16174},{15733,16193},{15868,22767},{15950,22836},{16552,17747},{16842,19855},{17514,19858},{17549,20066},{17577,20060},{17597,24159},{17605,21077},{17724,24160}.
= complement of X(3871)
= midpoint of X(i) and X(j) for these {i,j}: {{3585, 5288}, {4861, 5086}}.
= reflection of X(i) in X(j) for these {i,j}: {{35, 4999}, {15338, 5267}, {20612, 942}}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2886, 442), (2, 5082, 5687), (5, 8, 17757), (8, 5603, 5730), (8, 11680, 5), (10, 11, 4187), (10, 10914, 1145), (10, 12053, 392), (10, 21630, 3884), (21, 149, 15171), (145, 2476, 495), (377, 10529, 999), (497, 19843, 405), (499, 1376, 13747), (946, 4847, 72), (958, 1479, 11113), (958, 11235, 1479), (1125, 3925, 17529), (1329, 3829, 7741), (1329, 7741, 17533), (1698, 3816, 17575), (2550, 3086, 474), (2551, 10591, 17556), (2886, 3813, 1), (3006, 3702, 3695), (3244, 3822, 15888), (3434, 10527, 3), (3617, 4193, 3820), (3626, 3814, 21031), (3632, 7951, 12607), (3679, 3829, 17533), (3679, 7741, 1329), (3816, 9710, 1698), (3820, 10593, 4193), (4863, 11375, 3811), (6675, 15172, 1621), (6980, 12645, 10942), (7173, 21031, 3814), (9669, 9708, 2478), (12608, 14872, 13257), (17046, 20257, 4904).
= complement of the isogonal of X(20615).
= X(i)-complementary conjugate of X(j) for these (i,j): {{56, 4075}, {596, 1329}, {20615, 10}}.
-----------------------------------------------------------
= complement of X(3871)
= midpoint of X(i) and X(j) for these {i,j}: {{3585, 5288}, {4861, 5086}}.
= reflection of X(i) in X(j) for these {i,j}: {{35, 4999}, {15338, 5267}, {20612, 942}}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2886, 442), (2, 5082, 5687), (5, 8, 17757), (8, 5603, 5730), (8, 11680, 5), (10, 11, 4187), (10, 10914, 1145), (10, 12053, 392), (10, 21630, 3884), (21, 149, 15171), (145, 2476, 495), (377, 10529, 999), (497, 19843, 405), (499, 1376, 13747), (946, 4847, 72), (958, 1479, 11113), (958, 11235, 1479), (1125, 3925, 17529), (1329, 3829, 7741), (1329, 7741, 17533), (1698, 3816, 17575), (2550, 3086, 474), (2551, 10591, 17556), (2886, 3813, 1), (3006, 3702, 3695), (3244, 3822, 15888), (3434, 10527, 3), (3617, 4193, 3820), (3626, 3814, 21031), (3632, 7951, 12607), (3679, 3829, 17533), (3679, 7741, 1329), (3816, 9710, 1698), (3820, 10593, 4193), (4863, 11375, 3811), (6675, 15172, 1621), (6980, 12645, 10942), (7173, 21031, 3814), (9669, 9708, 2478), (12608, 14872, 13257), (17046, 20257, 4904).
= complement of the isogonal of X(20615).
= X(i)-complementary conjugate of X(j) for these (i,j): {{56, 4075}, {596, 1329}, {20615, 10}}.
-----------------------------------------------------------
P = X(20);
= COMPLEMENT OF X(11523)
3 a^3 b-a^2 b^2-3 a b^3+b^4+3 a^3 c+2 a^2 b c-a b^2 c-a^2 c^2-a b c^2-2 b^2 c^2-3 a c^3+c^4 : :
= 3 X[165] - X[3189], X[20] - 3 X[3928], 3 X[5770] - 2 X[6705], 3 X[5657] - X[6765], 3 X[5709] - X[6869], 5 X[3522] - X[12536], 3 X[10] - 2 X[12607], 3 X[3928] + X[12625], X[6769] - 3 X[14647], 3 X[10175] - 2 X[21077], 3 X[10165] - 2 X[22836].
= lies on these lines: {1,5745},{2,3984},{3,519},{4,527},{8,57},{9,938},{10,141},{11,3962},{20,3928},{40,5768},{56,6737},{63,950},{65,4847},{72,1210},{78,3911},{144,10392},{145,3601},{165,3189},{200,1467},{210,8582},{226,2476},{329,9581},{341,4899},{354,21677},{355,2095},{377,553},{387,3946},{405,5325},{443,3679},{452,3929},{497,12526},{515,5709},{516,5787},{517,6245},{528,5493},{529,20420},{551,6675},{579,2321},{610,5839},{758,946},{908,5154},{912,6260},{936,6692},{958,6738},{960,11019},{986,3755},{1001,6744},{1040,15954},{1125,5791},{1329,21060},{1376,6743},{1385,5771},{1737,5904},{1770,4880},{1834,3663},{1837,12527},{1998,10393},{2093,5082},{2136,6764},{2323,3562},{2478,3951},{2550,3339},{2551,5223},{2886,3671},{2900,10884},{3008,17054},{3243,5775},{3244,5855},{3419,4292},{3485,5231},{3487,5705},{3522,12536},{3617,9776},{3626,5708},{3632,15803},{3730,21096},{3811,6684},{3813,4301},{3825,4127},{3869,12053},{3876,5316},{3894,13407},{3901,12047},{3916,4304},{3927,5722},{3940,6700},{4001,5016},{4067,21616},{4208,6173},{4298,5794},{4313,20008},{4314,4640},{4654,5177},{4667,5717},{4685,16056},{4711,10855},{4855,20013},{5044,9843},{5046,17781},{5128,17784},{5175,9579},{5220,18250},{5273,5436},{5435,5438},{5657,6765},{5690,9940},{5770,6705},{5777,7682},{5795,18391},{5805,5850},{5843,22792},{5854,13226},{6282,12245},{6763,10572},{6769,14647},{6824,13464},{6847,7982},{9943,15733},{10165,22836},{10175,21077},{10202,10915},{10477,20258},{10529,11682},{10578,18231},{11529,19843},{12832,14740},{14054,18389},{14986,15829},{15933,17558},{17449,23675},{21214,24216}.
= complement of X(11523).
= midpoint of X(i) and X(j) for these {i,j}: {{8, 6762}, {20, 12625}, {2136, 6764}, {12245, 12629}}.
= reflection of X(i) in X(j) for these {i,j}: {946, 10916}, {3244, 11260}, {3811, 6684}, {4301, 3813}, {5882, 8666}, {12437, 3}, {12635, 1125}, {12640, 11362}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 942, 142), (10, 3874, 21620), (63, 12649, 950), (72, 1210, 3452), (145, 5744, 3601), (1737, 5904, 21075), (3868, 6734, 226), (3927, 5722, 12572), (3928, 12625, 20), (5175, 9965, 9579), (5435, 20007, 5438), (5777, 7682, 9842), (5791, 15934, 1125), (6744, 18249, 1001).
-----------------------------------------------------------
= complement of X(11523).
= midpoint of X(i) and X(j) for these {i,j}: {{8, 6762}, {20, 12625}, {2136, 6764}, {12245, 12629}}.
= reflection of X(i) in X(j) for these {i,j}: {946, 10916}, {3244, 11260}, {3811, 6684}, {4301, 3813}, {5882, 8666}, {12437, 3}, {12635, 1125}, {12640, 11362}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 942, 142), (10, 3874, 21620), (63, 12649, 950), (72, 1210, 3452), (145, 5744, 3601), (1737, 5904, 21075), (3868, 6734, 226), (3927, 5722, 12572), (3928, 12625, 20), (5175, 9965, 9579), (5435, 20007, 5438), (5777, 7682, 9842), (5791, 15934, 1125), (6744, 18249, 1001).
-----------------------------------------------------------
P = X(210);
= X(1)X(442)∩X(2)X(3158)
(a-b-c) (a^2-a b+2 b^2-a c-4 b c+2 c^2) : :
= 4 X[10] - X[2136], 4 X[2886] - X[2900], 4 X[1125] - X[3189], 2 X[8] + X[3680], X[1] - 4 X[3813], 5 X[1698] - 2 X[3913], X[9] + 2 X[6601], 2 X[4] + X[6762], 5 X[3091] + X[6764], 4 X[5] - X[6765], 4 X[6666] - X[7674], 4 X[3829] - 3 X[7988], 2 X[3811] - 5 X[8227], X[3632] + 2 X[10912], X[40] - 4 X[10916], 4 X[946] - X[11523], 5 X[3616] - 2 X[12437], X[5691] + 2 X[12513], 7 X[3622] - X[12536], 5 X[3617] + X[12541], 7 X[7989] - 4 X[12607], 2 X[1] + X[12625], 8 X[3813] + X[12625], 2 X[355] + X[12629], 7 X[9780] - X[12632], 5 X[11522] - 2 X[12635], 5 X[3617] - 2 X[12640], X[12541] + 2 X[12640], 4 X[3036] - X[12641], X[1768] + 2 X[13271], X[11531] - 4 X[13463], 2 X[3174] - 5 X[20195], X[3680] - 4 X[21627], X[8] + 2 X[21627], 7 X[9624] - 4 X[22836].
= lies on these lines: {1,442},{2,3158},{4,6762},{5,6765},{8,3452},{9,497},{10,1058},{11,200},{40,6899},{55,5231},{57,3434},{63,149},{72,9614},{120,5272},{142,10580},{145,5226},{165,528},{210,11238},{226,3243},{312,4901},{354,6173},{355,7956},{390,5745},{392,3679},{496,936},{516,3928},{518,1699},{519,3545},{521,11193},{522,6545},{527,9812},{674,10439},{748,3939},{946,11523},{956,3586},{1024,2319},{1125,3189},{1146,5574},{1210,1706},{1329,4882},{1420,10529},{1697,6734},{1698,3913},{1738,5573},{1768,13271},{1837,4853},{2481,20935},{2550,5437},{2898,9436},{2999,17721},{3036,4900},{3058,4512},{3086,5438},{3091,6764},{3174,3925},{3295,5705},{3340,12649},{3555,9612},{3601,10527},{3616,12437},{3617,12541},{3622,12536},{3632,5087},{3633,10827},{3677,3914},{3681,10707},{3703,4873},{3705,3886},{3706,4007},{3756,8056},{3811,8227},{3816,8580},{3829,7988},{3870,5219},{3872,5727},{3873,4654},{3875,7179},{3911,17784},{3940,7743},{3944,16496},{3966,4034},{4323,20008},{4326,6067},{4373,16078},{4423,6600},{4514,11679},{4677,5854},{4859,21949},{4862,21342},{5049,17528},{5123,8168},{5175,10106},{5178,19861},{5225,12527},{5261,9797},{5268,24217},{5269,11269},{5436,19843},{5691,12513},{5722,9623},{5748,20015},{5784,12915},{5791,15172},{5837,9785},{5855,11224},{5880,10980},{6666,7674},{6745,10589},{7082,13274},{7174,24210},{7989,12607},{8055,10005},{9624,22836},{9708,18527},{9780,12632},{10453,17296},{10591,21075},{11522,12635},{11531,13463},{12526,12701},{16572,21073},{17284,19589},{17597,23681},{21740,22837}.
= reflection of X(i) in X(j) for these {i,j}: {1699, 11235}, {3158, 2}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 5274, 3452), (8, 6557, 6555), (8, 12053, 15829), (8, 21627, 3680), (11, 4863, 200), (63, 149, 9580), (497, 4847, 9), (1210, 5082, 1706), (2550, 11019, 5437), (3617, 12541, 12640), (3870, 11680, 5219), (3925, 10582, 20195), (8056, 21267, 3756).
= X(8)-waw conjugate of X(9).
= X(6558)-Ceva conjugate of X(522).
= crosspoint of X(8) and X(4373).
= crosssum of X(i) and X(j) for these (i,j): {{56, 3052}, {604, 21059}}.
= barycentric product X(i)X(j) for these {i,j}: {{8, 4859}, {333, 21949}}.
= barycentric quotient X(i)/X(j) for these {i,j}: {{4859, 7}, {21949, 226}}.
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= reflection of X(i) in X(j) for these {i,j}: {1699, 11235}, {3158, 2}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 5274, 3452), (8, 6557, 6555), (8, 12053, 15829), (8, 21627, 3680), (11, 4863, 200), (63, 149, 9580), (497, 4847, 9), (1210, 5082, 1706), (2550, 11019, 5437), (3617, 12541, 12640), (3870, 11680, 5219), (3925, 10582, 20195), (8056, 21267, 3756).
= X(8)-waw conjugate of X(9).
= X(6558)-Ceva conjugate of X(522).
= crosspoint of X(8) and X(4373).
= crosssum of X(i) and X(j) for these (i,j): {{56, 3052}, {604, 21059}}.
= barycentric product X(i)X(j) for these {i,j}: {{8, 4859}, {333, 21949}}.
= barycentric quotient X(i)/X(j) for these {i,j}: {{4859, 7}, {21949, 226}}.
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P = X(390);
= COMPLEMENT OF X(3243)
(a-b-c) (3 a b-b^2+3 a c+2 b c-c^2) : :
= lies on these lines: {1,4878},{2,3243},{7,3617},{8,9},{10,141},{11,210},{55,5325},{75,4899},{144,4678},{145,18230},{200,5218},{226,3681},{333,7256},{355,382},{480,6737},{515,3358},{519,1001},{527,1478},{528,4669},{551,15570},{944,21153},{946,3678},{952,6594},{958,6600},{960,4342},{971,5690},{984,3755},{1210,3697},{1445,10106},{1788,4321},{2346,5260},{2551,4866},{2886,21060},{3008,3242},{3036,5856},{3057,7064},{3059,4111},{3158,5273},{3174,4882},{3189,5234},{3219,20095},{3244,4974},{3340,8232},{3621,8236},{3625,6541},{3671,9710},{3703,4061},{3706,4082},{3711,6745},{3715,4863},{3740,11019},{3751,4667},{3786,17197},{3876,12053},{3913,18249},{3929,17784},{3946,7174},{3983,8582},{3984,11526},{4015,10916},{4133,4439},{4310,17067},{4314,5302},{4344,16670},{4345,15829},{4349,4663},{4661,5249},{4668,5698},{4690,5845},{4691,5850},{4710,4738},{4711,15733},{4864,17337},{4915,8275},{4929,16833},{5252,12573},{5534,5770},{5657,5732},{5790,5805},{5806,12599},{5817,12245},{5904,11551},{6067,21031},{6692,8580},{6734,7705},{7951,21075},{8256,15587},{9053,17348},{9623,11041},{9668,12572},{9780,11038},{9948,11495},{9956,20330},{10039,18412},{10164,13226},{10389,20015},{10573,15298},{10624,15650},{10950,15837},{12447,12513},{12527,12943},{12564,12855},{12647,15299},{18357,18482},{21342,24175}.
= complement of X(3243).
= midpoint of X(i) and X(j) for these {i,j}: {8, 9}, {2550, 5223}, {7674, 12625}.
= reflection of X(i) in X(j) for these {i,j}: {{1, 6666}, {142, 10}, {5542, 3826}, {12437, 6600}, {18482, 18357}, {20330, 9956}}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 2321, 4923), (8, 3717, 2321), (8, 5686, 9), (8, 5837, 12640), (8, 10005, 4901), (10, 5542, 3826), (10, 21255, 3823), (210, 4847, 3452), (958, 6743, 12437), (1697, 5809, 15006), (3679, 5223, 2550), (3703, 4113, 4061), (3706, 4126, 4082), (3826, 5542, 142), (9780, 11038, 20195).
= X(8)-beth conjugate of X(142).
= crosssum of X(56) and X(1471).
= barycentric product X(4924)X(6557).
= barycentric quotient X(4924)/X(5435).
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= complement of X(3243).
= midpoint of X(i) and X(j) for these {i,j}: {8, 9}, {2550, 5223}, {7674, 12625}.
= reflection of X(i) in X(j) for these {i,j}: {{1, 6666}, {142, 10}, {5542, 3826}, {12437, 6600}, {18482, 18357}, {20330, 9956}}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 2321, 4923), (8, 3717, 2321), (8, 5686, 9), (8, 5837, 12640), (8, 10005, 4901), (10, 5542, 3826), (10, 21255, 3823), (210, 4847, 3452), (958, 6743, 12437), (1697, 5809, 15006), (3679, 5223, 2550), (3703, 4113, 4061), (3706, 4126, 4082), (3826, 5542, 142), (9780, 11038, 20195).
= X(8)-beth conjugate of X(142).
= crosssum of X(56) and X(1471).
= barycentric product X(4924)X(6557).
= barycentric quotient X(4924)/X(5435).
-----------------------------------------------------------
P = X(524);
= X(1)X(17959)∩X(30)X(511)
a (a-b-c) (b+c) (a^2+b^2-3 b c+c^2) : :
= lies on these lines: {1,17959},{30,511},{37,4097},{210,2321},{612,1962},{2136,2292},{2294,3174},{2643,19589},{2650,3680},{3169,3728},{3678,4527},{3686,11997},{3742,3946},{3743,3913},{3747,3939},{3753,3755},{3754,4743},{3773,3956},{3786,3877},{3794,4483},{3873,3875},{3919,4780},{3943,21889},{3950,21867},{3968,4085},{4015,4535},{4068,6600},{4133,4134},{4433,4516},{4512,4877},{6601,18698},{10158,10180},{12541,17164},{17163,18697},{21871,22312}.
= barycentric product X(i)X(j) for these {i,j}: {{8, 16611}, {9, 4442}, {2321, 7292}, {3701, 16784}, {7017, 23230}}.
= barycentric quotient X(i)/X(j) for these {i,j}: {{2832, 17096}, {4442, 85}, {6019, 16611}, {7292, 1434}, {16611, 7}, {16784, 1014}, {23230, 222}}.
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The locus as P moves along a line is another line.
= barycentric product X(i)X(j) for these {i,j}: {{8, 16611}, {9, 4442}, {2321, 7292}, {3701, 16784}, {7017, 23230}}.
= barycentric quotient X(i)/X(j) for these {i,j}: {{2832, 17096}, {4442, 85}, {6019, 16611}, {7292, 1434}, {16611, 7}, {16784, 1014}, {23230, 222}}.
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The locus as P moves along a line is another line.
Best regards,
Peter Moses.
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