HYACINTHOS 24851

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

Na, Nb, Nc =the NPC centers of IBC, ICA, IAB, resp.

A', B', C' = the reflections of I in BC, CA, AB, resp.

La, Lb, Lc = the reflections of A'Na, B'Nb, C'Nc in IA', IB', IC', resp.

1. La, Lb, Lc are concurrent

2. the parallels to La, Lb, Lc through A', B', C' are concurrent.

3. the parallels to La, Lb, Lc through Ia, Ib, Ic (where IaIbIc is the excentral triangle) are concurrent.

[Peter Moses]:

Hi Antreas,

 

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

on lines {{1,3},{8,4867},{10,5443},{12, 5844},{79,1320},{80,946},{145, 1478},{499,10595},{519,5086},{ 758,4861},{944,10483},{952, 3585},{958,3899},{1125,5330},{ 1483,7354},{1698,5289},{1731, 1953},{1770,5882},{1837,3656}, {1845,6198},{2779,7727},{3241, 4295},{3242,9047},{3583,10950} ,{3621,10590},{3623,4293},{ 3632,10827},{3633,9612},{3635, 4292},{3636,5442},{3679,5730}, {3869,5258},{3872,5904},{3877, 5259},{3878,5251},{3919,5253}, {3940,4668},{3970,4919},{4299, 7967},{4301,10572},{4677,4930} ,{4880,8666},{5270,10944},{ 5433,10283},{5444,6684},{5603, 7741},{7356,7979}}.

Reflection of X(i) in X(j) for these {i,j}: {{35, 1}, {5288, 4861}}.

3 X[35] - 4 X[2646], 3 X[1] - 2 X[2646].

(4r + R)X[1] - 2 r X[3].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,65,5563),(1,484,1385),(1, 3336,1319),(1,3340,5902),(1, 5697,3746),(1,5903,36),(1, 7280,10246),(1,7982,5697),(1, 7991,3612),(56,10247,1),(65, 10222,1),(942,5048,1),(1389, 10698,946),(1482,2099,1),( 5603,10573,7741).

 

2) at 1).

 

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

on lines {{1,3},{4,7161},{8,191},{9, 5560},{10,3583},{11,5445},{63, 3632},{71,1731},{78,3899},{79, 495},{80,3467},{100,3878},{ 214,5330},{238,3987},{404, 3884},{498,962},{515,4324},{ 516,3585},{519,6763},{550, 5559},{595,4642},{758,3871},{ 944,1768},{946,6949},{1018, 3496},{1203,4646},{1320,5303}, {1334,5011},{1478,6361},{1479, 5657},{1621,3754},{1698,4193}, {1699,6941},{1737,4857},{1749, 5441},{1759,3208},{1770,5270}, {1776,4330},{1837,3654},{1900, 7713},{2779,9904},{2802,2975}, {2943,6127},{3218,3244},{3219, 3626},{3555,4880},{3633,3895}, {3679,5086},{3730,5540},{3751, 9047},{3753,5259},{3869,8715}, {3870,3901},{3880,3916},{3885, 8666},{3898,5253},{3913,5904}, {3918,5047},{3935,4067},{4063, 6161},{4294,10573},{4295, 10056},{4299,9778},{4325, 10106},{4333,9613},{4338,5290} ,{4421,5730},{4640,5258},{ 4861,5267},{5251,5836},{5252, 10483},{5432,5443},{5444,5901} ,{5506,9780},{5531,5693},{ 5561,5726},{5687,5692},{6192, 7150},{6932,9589},{6963,9588}, {7031,9620},{9580,10826},{ 9785,10072}}.

Reflection of X(i) in X(j) for these {i,j}: {{1, 35}, {3585, 10039}, {4861, 5267}, {5288, 3916}}.

3 X[1] - 4 X[2646], 3 X[35] - 2 X[2646], 3 X[3679] - 2 X[5086].

(2r - R)X[1] - 4 r X[3].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,40,484),(1,46,3337),(1,165, 7280),(1,484,3336),(1,5131,56) ,(3,5697,1),(36,3057,1),(40, 1697,46),(40,5119,1),(46,1697, 1),(46,3337,3336),(46,5119, 1697),(55,5903,1),(65,3746,1), (191,5541,8),(484,3337,46),( 1155,9957,5563),(1737,10624, 4857),(1759,3208,5525),(3057, 3579,36),(3245,3746,65),(3295, 5902,1),(3612,7982,1),(4424, 5255,1),(4640,10914,5258),( 5563,9957,1),(5690,6284,80).

 

Also the parallels to La, Lb, Lc through triangle T’s vertices  are concurrent for T

 

intouch: a (2 a-3 b-3 c) (a+b-c) (a-b+c)::
on lines {{1,3},{7,1392},{8,6933},{12, 519},...}.

first circum perp: a^2 (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c+a^3 b c-a^2 b^2 c-a b^3 c+2 b^4 c-2 a^3 c^2-a^2 b c^2+4 a b^2 c^2-b^3 c^2+2 a^2 c^3-a b c^3-b^2 c^3+a c^4+2 b c^4-c^5)
on lines {{1,3},{4,993},{5,5251},{8, 6796},{10,6905},{20,5450},{21, 946},...}.

second circum perp: X(35).

tangential mid arc: a ((a+b+c) (a^3-a^2 b-a b^2+b^3-a^2 c+a b c-b^2 c-a c^2-b c^2+c^3)+2 a b (2 a-3 b-3 c) c Sin[A/2]+2 c (a^3-a^2 b+2 a b^2-b^3+a^2 c+b^2 c-a c^2+b c^2-c^3) Sin[B/2]+2 b (a^3+a^2 b-a b^2-b^3-a^2 c+b^2 c+2 a c^2+b c^2-c^3) Sin[C/2])::
on lines {{35,8077},{517,8091},...}.

hexyl: a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+5 a^4 b c-5 a^3 b^2 c-3 a^2 b^3 c+7 a b^4 c-2 b^5 c-a^4 c^2-5 a^3 b c^2+12 a^2 b^2 c^2-5 a b^3 c^2-b^4 c^2+4 a^3 c^3-3 a^2 b c^3-5 a b^2 c^3+4 b^3 c^3-a^2 c^4+7 a b c^4-b^2 c^4-2 a c^5-2 b c^5+c^6)
on lines {{1,3},{8,6326},{10,6949},...} .

Conway: 3 a^4-a^3 b-2 a^2 b^2+a b^3-b^4-a^3 c-a^2 b c-2 a^2 c^2+2 b^2 c^2+a c^3-c^4::
on lines {{7,1392},{8,4640},{10,21},{ 20,145},...}

 

Best regards,

Peter Moses.