HYACINTHOS 29407

[Antreas P. Hatzipolskis]:

 

VARIATIONS of Hyacinthos 29402 

 

Let ABC be a triangle and P a point on the (I) [ = incircle]

Denote:

Ga, Gb, Gc = the G's (centroids) of PBC, PCA, PAB, resp.

The (I) [ = incircle] of GaGbGc passes through G of ABC.

Which is its center for P on the (I) of ABC:  X(11), ....  ?


[Peter Moses]:


Hi Antreas,

For a point P{u, v, w}, (a + b - c) (a - b + c) (v - w)^2 : : is on the incircle of ABC.
The radius of the incircle of GaGbGc = r / 3.
The center of the incircle of GaGbGc = 2 a (a-b-c) (b+c) u^2-2 (a-b-c) (a-b+c) (b+c) u v-(a-b+c) (a^2+2 a b+3 b^2+2 b c-c^2) v^2-2 (a-b-c) (a+b-c) (b+c) u w+2 (a+b-c) (a-b+c) (a+2 b+2 c) v w-(a+b-c) (a^2-b^2+2 a c+2 b c+3 c^2) w^2 : :

P = X(7) -> X(11) on the incircle -> 

 

= MIDPOINT OF X(80) AND X(15015) =

 

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

= X[1] + 2 X[3036],X[1] - 4 X[6667],7 X[2] - X[10031],4 X[5] - X[1537],X[8] + 2 X[1387],2 X[8] + X[25416],X[8] + 5 X[31272],X[8] + 3 X[32558],2 X[10] + X[11],4 X[10] - X[1145],X[10] + 2 X[6702],10 X[10] - X[13996],5 X[10] + X[21630],2 X[11] + X[1145],X[11] - 4 X[6702],5 X[11] + X[13996],5 X[11] - 2 X[21630],X[65] + 2 X[18254],X[72] + 2 X[12736],X[80] + 5 X[1698],X[80] + 2 X[3035],2 X[80] + X[10609],X[100] - 7 X[9780],X[100] + 2 X[12019],2 X[100] + X[12690],X[104] + 5 X[5818],X[119] - 4 X[9956],X[119] + 2 X[12619],4 X[119] - X[13257],2 X[149] + X[12732],X[153] + 2 X[13226],X[214] - 4 X[3634],2 X[214] - 5 X[31235],X[355] + 2 X[6713],4 X[1125] - X[1317],2 X[1125] + X[15863],X[1145] + 8 X[6702],5 X[1145] - 2 X[13996],5 X[1145] + 4 X[21630],X[1317] + 2 X[15863],X[1320] + 5 X[3617],4 X[1387] - X[25416],2 X[1387] - 5 X[31272],2 X[1387] - 3 X[32558],5 X[1656] - 2 X[11729],5 X[1656] + X[19914],5 X[1698] - 2 X[3035],10 X[1698] - X[10609],5 X[1698] - X[15015],X[1737] + 2 X[5123],2 X[1737] + X[17757],4 X[3035] - X[10609],X[3036] + 2 X[6667],7 X[3090] - X[10698],X[3555] - 4 X[18240],5 X[3616] + X[12531],5 X[3616] - 2 X[12735],7 X[3624] - X[7972],X[3626] + 2 X[33709],4 X[3628] - X[19907],8 X[3634] - 5 X[31235],5 X[3697] - 2 X[14740],5 X[3698] + X[17638],4 X[3812] - X[11570],4 X[3822] - X[12831],4 X[3826] - X[10427],4 X[3828] - X[6174],X[4511] + 2 X[11545],5 X[4668] + X[26726],2 X[5044] + X[6797],2 X[5083] - 5 X[5439],4 X[5123] - X[17757],X[5176] + 2 X[15325],2 X[5836] + X[12758],X[6224] - 13 X[19877],X[6246] + 2 X[6684],2 X[6246] + X[24466],X[6326] - 4 X[20400],4 X[6684] - X[24466],20 X[6702] + X[13996],10 X[6702] - X[21630],4 X[6723] - X[31525],7 X[9780] + 2 X[12019],14 X[9780] + X[12690],2 X[9956] + X[12619],16 X[9956] - X[13257],X[10265] + 5 X[31399],X[10914] + 2 X[15558],X[11362] + 2 X[16174],2 X[11729] + X[19914],4 X[12019] - X[12690],X[12119] - 7 X[31423],X[12531] + 2 X[12735],8 X[12619] + X[13257],X[12751] + 2 X[20418],X[13996] + 2 X[21630],X[14872] + 2 X[15528],5 X[15017] - 17 X[30315],X[17636] + 5 X[25917],5 X[18230] + X[20119],4 X[19878] - X[33812],X[25416] - 10 X[31272],X[25416] - 6 X[32558],5 X[31272] - 3 X[32558]

= lies on these lines: {1,3036}, {2,952}, {5,1537}, {8,1387}, {10,11}, {12,5883}, {21,33814}, {65,18254}, {72,12736}, {80,1698}, {100,405}, {104,474}, {119,125}, {149,5084}, {153,443}, {214,3634}, {355,6713}, {377,10742}, {404,18357}, {406,1862}, {452,13199}, {475,12138}, {515,21154}, {517,17533}, {518,1737}, {519,32557}, {528,19875}, {632,3897}, {900,14431}, {958,10090}, {1001,10087}, {1125,1317}, {1320,3617}, {1329,5692}, {1376,10058}, {1482,6931}, {1484,17527}, {1656,5554}, {1772,24433}, {1788,24465}, {2475,22799}, {2478,10738}, {2800,3753}, {2804,14429}, {2829,5587}, {3090,10698}, {3555,18240}, {3614,3754}, {3616,12531}, {3624,7972}, {3626,33709}, {3628,19907}, {3679,5854}, {3697,14740}, {3698,17638}, {3756,24222}, {3812,11570}, {3813,15079}, {3816,5533}, {3822,12831}, {3826,10427}, {3828,6174}, {3847,5697}, {4193,5690}, {4205,9978}, {4208,13243}, {4511,11545}, {4668,26726}, {4857,32157}, {4881,28224}, {4996,5260}, {5044,6797}, {5046,22938}, {5083,5439}, {5129,20095}, {5151,11105}, {5154,22791}, {5176,15325}, {5187,12702}, {5541,31435}, {5657,17556}, {5817,17532}, {5836,12758}, {5840,11113},{6224,19877}, {6246,6684}, {6264,8583}, {6265,19860}, {6326,20400}, {6723,31525}, {6788,17724}, {6831,32554}, {6856,9952}, {6857,9945}, {6904,12248}, {6910,12747}, {6913,12775}, {6921,18525}, {7741,8256}, {8165,15650}, {8582,10265}, {8728,11698}, {8988,13936}, {9809,11024}, {9963,17558}, {10039,10179}, {10074,25524}, {10592,12532}, {10593,14923}, {10728,26062},{10956,20118},{11108,12331},{11362,16174},{11715,17614},{12119,31423},{12737,19861},{12751,20418},{12773,16408},{13587,28186},{13883,13976},{13893,19077},{13947,19078},{14193,26073},{14439,21044},{14872,15528},{15017,30315},{17100,19525}, {17575,24987}, {17768,31160}, {18230,20119}, {19878,33812}, {25490,26029}, {25491,26030}, {25513,26046}

= midpoint of X(i) and X(j) for these {i,j}: {80, 15015}, {3679, 16173}

= reflection of X(i) in X(j) for these {i,j}: {10609, 15015}, {15015, 3035}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):  {8, 1387, 25416}, {8, 31272, 1387}, {10, 11, 1145}, {10, 6702, 11}, {10, 17606, 24390}, {10, 17619, 4187}, {11, 13996, 21630}, {80, 1698, 3035}, {80, 3035, 10609}, {100, 12019, 12690}, {214, 3634, 31235}, {1125, 15863, 1317}, {1656, 19914, 11729}, {1737, 5123, 17757}, {3036, 6667, 1}, {3616, 12531, 12735}, {3753, 10175, 17530}, {6246, 6684, 24466}, {7705, 25005, 5}, {9956, 12619, 119}, {9956, 24982, 442}


P = X(514) -> X(1317) on the incircle -> 

= X(1)X(1145)∩X(2)X(292) =

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

= 2 X[1] + X[1145],X[1] + 2 X[3035],4 X[1] - X[25416],7 X[1] - X[26726],5 X[2] + X[10031],2 X[3] + X[1537],X[3] + 2 X[11729],X[8] + 2 X[12735],2 X[10] + X[1317],2 X[10] - 5 X[31235],X[11] + 2 X[214],X[11] - 4 X[1125],2 X[11] + X[10609],4 X[11] - X[12690],5 X[11] - 8 X[33709],X[72] + 2 X[5083],X[80] - 7 X[3624],X[80] - 4 X[6667],X[100] + 2 X[1387],X[100] + 5 X[3616],5 X[100] + X[9802],4 X[100] - X[12732],2 X[104] + X[13257],X[119] + 2 X[1385],2 X[140] + X[19907],X[149] + 2 X[9945],X[214] + 2 X[1125],4 X[214] - X[10609],8 X[214] + X[12690],5 X[214] + 4 X[33709],2 X[551] + X[6174],5 X[631] + X[10698],X[908] + 2 X[5126],2 X[946] + X[24466],2 X[960] + X[11570],2 X[993] + X[12831],2 X[1001] + X[10427],8 X[1125] + X[10609],16 X[1125] - X[12690],5 X[1125] - 2 X[33709],X[1145] - 4 X[3035],2 X[1145] + X[25416],7 X[1145] + 2 X[26726],X[1317] + 5 X[31235],2 X[1319] + X[17757],X[1320] - 7 X[3622],2 X[1387] - 5 X[3616],10 X[1387] - X[9802],8 X[1387] + X[12732],X[1537] - 4 X[11729],5 X[1698] - 2 X[3036],5 X[1698] + X[7972],8 X[3035] + X[25416],14 X[3035] + X[26726],2 X[3036] + X[7972],7 X[3526] - X[19914],X[3555] + 2 X[14740],25 X[3616] - X[9802],20 X[3616] + X[12732],7 X[3624] - 4 X[6667],4 X[3634] - X[15863],2 X[3634] + X[33812],8 X[3636] + X[13996],2 X[3828] + X[11274],X[4511] + 2 X[15325],2 X[5087] + X[21578],5 X[5439] - 2 X[12736],2 X[5542] + X[6068],11 X[5550] + X[6224],11 X[5550] - 2 X[12019],11 X[5550] - 5 X[31272],2 X[5901] + X[33814],2 X[5972] + X[31525],X[6154] + 14 X[15808],X[6154] + 2 X[21630],X[6224] + 2 X[12019],X[6224] + 5 X[31272],X[6265] + 2 X[6713],X[6326] + 2 X[20418],2 X[6684] + X[25485],2 X[6702] - 5 X[19862],2 X[6702] + X[33337],X[6735] + 2 X[25405],5 X[8227] + X[12119],7 X[9624] - X[14217],7 X[9780] - X[12531],4 X[9802] + 5 X[12732],2 X[10609] + X[12690],X[10609] + 4 X[32557],5 X[10609] + 16 X[33709],X[10707] - 3 X[32558],2 X[11813] + X[15326],2 X[12019] - 5 X[31272],X[12611] + 2 X[13624],X[12665] + 2 X[12675],X[12690] - 8 X[32557],5 X[12690] - 32 X[33709],X[12743] + 2 X[17647],X[12751] - 4 X[20400],X[12832] + 2 X[30144],X[15015] + 3 X[25055],5 X[15017] + 7 X[30389],2 X[15254] + X[25558],7 X[15808] - X[21630],X[15863] + 2 X[33812],X[16173] - 3 X[25055],X[17660] + 2 X[18254],X[17660] + 5 X[25917],2 X[18254] - 5 X[25917],5 X[19862] + X[33337],7 X[25416] - 4 X[26726],5 X[32557] - 4 X[33709]

= lies on these lines: {1,1145}, {2,952}, {3,1537}, {8,12735}, {10,1317}, {11,214}, {21,33860}, {30,4881}, {36,17768}, {72,5083}, {80,3624}, {100,474}, {104,405}, {106,17724}, {119,1385}, {140,19907}, {149,443}, {153,5084}, {377,10738}, {392,2800}, {404,5901}, {406,12138}, {452,12248}, {475,1862}, {515,17533}, {528,15015}, {549,3877}, {551,2802}, {631,10698}, {758,5298}, {900,14419}, {908,5126}, {946,24466}, {954,5856}, {958,10074}, {960,11570}, {962,19537}, {993,12831}, {997,12739}, {1001,10058}, {1319,10956}, {1320,3622}, {1329,21842}, {1375,24559}, {1376,10087}, {1388,26364}, {1482,6921}, {1483,25005}, {1484,8728}, {1621,17100}, {1698,3036}, {1768,31435}, {2475,22938}, {2478,10742}, {2771,5642}, {2804,11125}, {2829,3576}, {2886,5533}, {3109,25533}, {3303,25438}, {3485,24465}, {3526,19914}, {3555,14740}, {3634,15863}, {3636,13996}, {3828,11274}, {3897,11698}, {3898,4995}, {4188,22791}, {4190,18493}, {4208,9963}, {4511,15325}, {4855,11373}, {4996,5253}, {5046,22799}, {5087,21578}, {5250,12515}, {5433,12832}, {5439,12736}, {5440,5853}, {5444,6690}, {5542,6068}, {5550,6224}, {5603,16371}, {5686,14151}, {5690,17566}, {5730,7288}, {5731,17556}, {5794,10073}, {5840,5886}, {5882,17619}, {5972,31525}, {6154,15808}, {6265,6713}, {6326,8583}, {6684,25485}, {6702,19862}, {6735,25405}, {6767,13278}, {6857,13226}, {6904,13199}, 6931,18525}, {6933,12747}, {7968,13922}, {7969,13991}, {8068,25466}, {8227,12119}, {9155,9978}, {9624,14217}, {9778,19705}, {9780,12531}, {9809,19526}, {10090,11507}, {10176,31157}, {10283,17564}, {10707,32558}, {10728,26129}, {11108,12773}, {11230,17530}, {11715,17575}, {11813,15326}, {12331,16408}, {12611,13624}, {12619,24987}, {12665,12675}, {12737,19860}, {12738,17590}, {12751,20400}, {13243,17558}, {13279,16410}, {13587,28174}, {13902,19112}, {13959,19113}, {15017,25522}, {15178,24982}, {15251,16377}, {15254,25558}, {16203,25875}, {16370,21151}, {16383,28915}, {17044,25532}, {17529,22935}, {17580,20095}, {17660,18254}, {18224,22766}, {18861,19525},{24928,27385}

= midpoint of X(i) and X(j) for these {i,j}: {214, 32557}, {5686, 14151}, {15015, 16173}
= reflection of X(i) in X(j) for these {i,j}: {11, 32557}, {21154, 10165}, {23513, 11230}, {32557, 1125}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1145, 25416}, {1, 3035, 1145}, {3, 11729, 1537}, {11, 214, 10609}, {11, 10609, 12690}, {80, 3624, 6667}, {100, 3616, 1387}, {214, 1125, 11}, {1125, 17614, 442}, {1317, 31235, 10}, {1698, 7972, 3036}, {3634, 33812, 15863}, {5550, 6224, 31272}, {6224, 31272, 12019}, {15015, 25055, 16173}, {17660, 25917, 18254}, {19862, 33337, 6702}

P = X(1) -> X(1358) on the incircle -> 

= X(2)X(28915)∩X(10)X(1358) =

= 2*a^4*b - 5*a^3*b^2 + 3*a^2*b^3 - 3*a*b^4 + 3*b^5 + 2*a^4*c - 6*a^3*b*c + 7*a^2*b^2*c - 6*a*b^3*c - 5*b^4*c - 5*a^3*c^2 + 7*a^2*b*c^2 + 10*a*b^2*c^2 + 2*b^3*c^2 + 3*a^2*c^3 - 6*a*b*c^3 + 2*b^2*c^3 - 3*a*c^4 - 5*b*c^4 + 3*c^5 : :

= 2 X[10] + X[1358],4 X[1125] - X[3021],5 X[1698] - 2 X[3039]

= lies on these lines: {2,28915}, {10,1358}, {105,474}, {115,120}, {377,10743}, {405,1292}, {443,20344}, {528,15015}, {1125,3021}, {1698,3039}, {2478,15521}, {2809,3753}, {4187,5511}, {6714,13747}, {11716,17614}, {17580,20097}

Best regards,
Peter Moses.