HYACINTHOS 28407

[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 =: PA', PB', PC', resp
 
Ra, Rb, Rc =  the reflections of R1, R2, R3 in B1C1, C1A1, A1B1, resp.

R'1, R'2, R'3 = the reflections of Ra, Rb, Rc in AI, BI, CI, resp.

R'1, R'2, R'3 are concurrent at a point P'

Which is it in terms of P?

We have here a conjugacy:

(P')' = P

ie

Denote:

R'a, R'b, R'c = the reflections of R'1, R'2, R'3 in B1C1, C1A1, A1B1. resp.
R"1, R"2, R"3 = the reflections of R'a, R'b, R'c in AI, BI, CI, resp.

We have R"1, R"2, R"3 = R1, R2, R3, resp. (concurrent at P)

[Peter Moses]:

 

Hi Antreas,  

>Which is it in terms of P?

a((b+c) (a^2-b^2-c^2) p+(a^2 b-b^3+a^2 c+2 a b c+b^2 c+b c^2-c^3) q+(a^2 b-b^3+a^2 c+2 a b c+b^2 c+b c^2-c^3) r) : :

Examples:

{1,65}
{4,1071}
{7,5728}
{8,3555}
{10,3874}
{11,11570}
{30,30}
{65,1}
{72,3868}
{79,17637}
{80,17660}
{145,10914}
{210,3894}
{222,1905}
{226,18389}
{354,5902}

P = X(2); 

=  X(1)X(3052)∩X(2)X(72)

a (3 a^2 b-3 b^3+3 a^2 c+4 a b c+b^2 c+b c^2-3 c^3) : : 

= lies on these lines: {1,3052},{2,72},{3,11520},{7,3419},{8,4004},{9,17542},{30,1071},{46,4421},{56,12559},{57,5440},{63,15934},{65,519},{78,5708},{210,3828},{226,17530},{354,392},{376,517},{381,912},{405,3929},{474,11523},{518,599},{527,5728},{528,11570},{537,21080},{549,10202},{597,9021},{956,11529},{960,3901},{971,3543},{995,3999},{997,4860},{999,4930},{1125,3962},{1159,3872},{1210,17533},{1385,16139},{1698,4533},{1739,4849},{1858,11238},{2093,3243},{2095,18446},{2771,10706},{2796,12723},{2802,4744},{2886,11551},{3057,3881},{3058,12711},{3218,17549},{3219,16861},{3244,4757},{3306,3940},{3333,5730},{3338,12635},{3339,5687},{3488,9965},{3524,9940},{3534,13369},{3545,5777},{3601,19704},{3626,3922},{3632,10107},{3634,4005},{3649,10916},{3653,13373},{3654,11239},{3656,11240},{3681,3921},{3697,3812},{3698,4745},{3740,4539},{3742,5692},{3754,4669},{3811,5221},{3829,12047},{3833,4134},{3839,5806},{3869,5045},{3877,5049},{3878,17609},{3889,9957},{3892,5919},{3899,10179},{3902,17145},{3927,16857},{3951,11108},{3984,16408},{4067,19883},{4127,19862},{4234,5208},{4654,14054},{4677,5836},{4737,20892},{4906,5315},{4980,5295},{5057,18527},{5122,23958},{5563,16126},{5570,10072},{5693,13374},{5722,5905},{5844,10273},{5887,6583},{6147,6734},{6875,15178},{6906,10222},{7373,11682},{7672,18419},{7682,13257},{9841,11531},{10056,13750},{10122,17525},{10156,15721},{10246,21165},{11111,15933},{11220,15683},{11227,15692},{12005,14110},{12109,21849},{12527,17706},{14450,22793},{15650,19536},{15803,19705},{16465,17579},{17619,21077},{19251,22345},{21969,23154}.

P = X(3); 

=  X(1)X(3)∩X(2)X(5761)

a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c-2 a^4 b c-a b^4 c+2 b^5 c-a^4 c^2+b^4 c^2-2 a^3 c^3-4 b^3 c^3+2 a^2 c^4-a b c^4+b^2 c^4+a c^5+2 b c^5-c^6) : : 

= lies on these lines: {1,3},{2,5761},{4,912},{5,72},{7,6850},{8,6826},{10,6881},{19,3211},{20,13369},{28,110},{30,1071},{34,3157},{37,5755},{52,1866},{63,3560},{68,5130},{78,6911},{119,21077},{140,5439},{142,3754},{155,1829},{210,9956},{225,1830},{226,6842},{227,5399},{283,18180},{329,5804},{355,518},{381,5715},{382,971},{389,12109},{392,5771},{429,12259},{443,10597},{500,15852},{511,13408},{515,3874},{516,5884},{546,5927},{550,10167},{579,8609},{758,946},{916,12162},{938,5758},{944,3873},{950,7491},{952,3555},{960,5791},{962,5768},{1046,3073},{1064,2650},{1066,1254},{1104,5398},{1210,6882},{1320,12776},{1389,4861},{1393,22350},{1439,22464},{1479,1858},{1512,10942},{1656,5044},{1699,3901},{1770,5840},{1828,5446},{1836,10525},{1837,10526},{1870,3562},{1871,15762},{1891,12134},{1898,3583},{1902,11472},{1998,19541},{2262,2323},{2771,7728},{2778,12262},{2800,4084},{2802,9946},{2808,13474},{2836,9970},{3090,3876},{3218,6906},{3219,6920},{3419,6917},{3487,6825},{3488,6868},{3529,11220},{3577,6762},{3649,15908},{3651,18444},{3678,10175},{3681,5818},{3753,5690},{3811,11499},{3812,10198},{3817,4067},{3827,19149},{3851,10157},{3869,5603},{3877,6857},{3878,5745},{3881,5882},{3889,7967},{3892,13607},{3894,5691},{3916,6914},{3927,6913},{3940,6918},{3962,5694},{4018,8727},{4185,9928},{4219,15062},{4297,12005},{4311,5083},{4325,12119},{4463,5797},{4848,12736},{4857,16155},{5231,18493},{5295,20237},{5435,6961},{5440,6924},{5480,9021},{5497,6011},{5534,18518},{5587,5904},{5657,6989},{5692,8227},{5703,6954},{5704,6978},{5713,5752},{5714,6982},{5720,11523},{5722,5812},{5728,5762},{5734,5744},{5748,6981},{5763,6922},{5770,6847},{5787,5878},{5841,10572},{5844,10914},{5883,6684},{6147,6907},{6261,12559},{6288,14872},{6797,19914},{6863,11374},{6885,14923},{6908,11036},{6985,11520},{7970,13190},{7978,13218},{7983,12190},{7984,12382},{8261,16139},{9581,18397},{9856,18544},{10156,15720},{10449,20928},{10531,11415},{10625,11573},{10693,12261},{10698,13279},{10705,13119},{11246,11826},{11571,12750},{12635,22753},{12650,12687},{12675,18481},{12695,13743},{12711,15171},{13099,13314},{13226,17654},{13729,17484},{17563,17612},{17625,18990},{17661,22799},{17857,18491},{18239,22792},{18254,23513},{18357,18908}.

P = X(5); 

=  X(1)X(1399)∩X(3)X(3218)

a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c-a^2 b^3 c-a b^4 c+b^5 c-a^4 c^2+b^4 c^2-2 a^3 c^3-a^2 b c^3-2 b^3 c^3+2 a^2 c^4-a b c^4+b^2 c^4+a c^5+b c^5-c^6) : : 

= lies on these lines: {1,1399},{3,3218},{4,17483},{5,226},{7,6917},{10,5885},{30,1071},{40,3894},{52,23154},{57,6924},{65,952},{72,140},{145,6948},{182,9021},{354,5887},{355,5270},{381,12528},{495,13750},{496,1858},{517,550},{518,5690},{548,10167},{549,9940},{632,5044},{758,1385},{916,5876},{938,6929},{946,1484},{960,13373},{971,3627},{1125,5694},{1482,3873},{1657,11220},{1735,5399},{1807,3075},{1870,23070},{2095,6985},{2167,19210},{2800,3881},{2801,18480},{3157,12161},{3337,6326},{3339,5534},{3487,5770},{3526,3876},{3555,5844},{3560,15934},{3562,18455},{3576,3901},{3628,5439},{3635,10284},{3670,5396},{3678,11231},{3845,5806},{3850,5927},{3869,10246},{3877,17571},{3878,15178},{3889,10247},{3916,7508},{3927,6883},{4067,10165},{4325,5903},{4430,12245},{4880,10902},{5045,10283},{5083,19907},{5208,15952},{5221,11499},{5446,12109},{5536,16132},{5563,6265},{5693,5886},{5708,6911},{5728,5843},{5730,16203},{5883,9956},{5904,15016},{5905,6928},{6001,22791},{6101,11573},{6824,11036},{6868,9965},{6885,21454},{6902,17484},{6907,14054},{6923,12649},{6936,20078},{6942,23958},{6955,20013},{7171,7982},{7330,11518},{10269,12635},{10273,10914},{10942,18838},{11009,11571},{11227,15712},{11230,20117},{11362,13145},{11567,11715},{12009,19862},{12558,22798},{12699,15071},{12711,15172},{13243,21669},{14872,18357},{15096,18406},{15931,16139},{21740,22765}.

P = X(6); 

=  X(1)X(159)∩X(6)X(169)

a (a^4 b-b^5+a^4 c+2 a b^3 c+b^4 c+2 a b c^3+b c^4-c^5) : : 

= lies on these lines: {1,159},{6,169},{7,8},{38,22097},{46,12329},{72,141},{81,105},{182,10202},{210,3844},{511,13408},{517,990},{611,13750},{613,5570},{912,1352},{1071,1503},{1172,18178},{1284,17447},{1738,21867},{1824,3782},{1843,23154},{1858,12589},{1876,6180},{2002,18838},{2352,18607},{2771,14982},{2809,3755},{2879,23770},{3057,3100},{3313,11573},{3555,5846},{3589,5439},{3619,3876},{3666,17441},{3744,20999},{3751,5902},{3763,5044},{3821,4523},{3873,19993},{3874,5847},{4267,18183},{4463,17184},{5085,9940},{5728,5845},{5777,10516},{5848,11570},{5903,16496},{9028,18389},{9053,10914},{9895,24159},{11220,14927},{12723,17768},{16475,18398}.

Best regards,
Peter Moses.