HYACINTHOS 29319

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

O' = the circumcenter of NaNbNc

The reflections of O'Na, O'Nb, O'Nc in PA, PB, PC, resp. are concurrent on the line at infinity (they are parallels).

Point of concurrence in terms of P?



[Peter Moses]:


Hi Antreas,

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

Some examples:
X(1) -->  X(517)
X(2) -->  X(3849)
X(3) -->  X(30)
X(5) -->  X(32744)
X(6) -->  X(8705)
X(13) -->  X(530)
X(14) -->  X(531)
X(74) -->  X(5663)
X(80) -->  X(952)
X(98) -->  X(2782)
X(99) -->  X(2782)
X(100) -->  X(952)
X(101) -->  X(2808)
X(102) -->  X(2818)
X(103) -->  X(2808)
X(104) -->  X(952)
X(105) -->  X(28915)
X(109) -->  X(2818)
X(110) -->  X(5663)
X(265) -->  X(5663)
X(476) -->  X(16168)
X(477) -->  X(16168)
X(485) -->  X(32436)
X(486) -->  X(32433)
X(671) -->  X(9830)
X(930) -->  X(25150)
X(1113) -->  X(30)
X(1114) -->  X(30)
X(1141) -->  X(25150)
X(1292) -->  X(28915)
X(1379) -->  X(511)
X(1380) -->  X(511)
X(1381) -->  X(517)
X(1382) -->  X(517)
...
...

----------------------------------------------

X(8) --> 

 

 X(1)X(5123) ∩ X(8)X(1319) =

 

= 4*a^4 - 5*a^3*b - 2*a^2*b^2 + 5*a*b^3 - 2*b^4 - 5*a^3*c + 14*a^2*b*c - 7*a*b^2*c - 2*a^2*c^2 - 7*a*b*c^2 + 4*b^2*c^2 + 5*a*c^3 - 2*c^4 : : 

= lies on these  lines: {1,5123},{8,1319},{10,25405},{30,511},{36,3632},{145,1837},{355,22835},{484,6762},{496,3244},{786,2823},{944,13528},{956,32760},{1155,3621},{1317,6735},{1376,5193},{1483,10915},{2077,3913},{2136,10085},{2781,23066},{3583,26726},{3625,5126},{3626,6681},{3633,9614},{3838,5252},{4511,20586},{4844,28191},{5057,20014},{5080,5225},{5122,24391},{5183,20053},{5440,7972},{5690,18857},{5836,10944},{5881,10912},{6700,33559},{8666,23961},{9422,28863},{10106,10107},{10950,17615},{11236,16200},{11256,12531},{11681,33176},{12629,17857},{12641,17613},{20054,20067},{24477,31145},{25416,30384},{27870,33899}

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X(15) --> 

 

 X(15)X(1337) ∩ X(30)X(511)  =

 

= a^2*(b^2*(a^2 - b^2)*(Sqrt[3]*c^2 + 2*S)*(Sqrt[3]*(a^2 + b^2 - c^2) + 2*S) + c^2*(a^2 - c^2)*(Sqrt[3]*b^2 + 2*S)*(Sqrt[3]*(a^2 - b^2 + c^2) + 2*S)) : :

= lies on these lines: {15,1337},{30,511},{51,15768},{396,15929},{621,10210},{623,33500},{2781,2823},{2979,16259},{3060,30439},{28175,28199},{6671,32223},{7927,28213},{8172,25177},{29122,29134},{14178,25225},{14538,18863},{16267,16461},{25219,25221}

----------------------------------------------

 

X(16) -->

 

 X(16)X(1338) ∩ X(30)X(511) =

 

 = a^2*(b^2*(a^2 - b^2)*(Sqrt[3]*c^2 - 2*S)*(Sqrt[3]*(a^2 + b^2 - c^2) - 2*S) + c^2*(a^2 - c^2)*(Sqrt[3]*b^2 - 2*S)*(Sqrt[3]*(a^2 - b^2 + c^2) - 2*S)) : :

= lies on these lines: {16,1338},{30,511},{51,15769},{395,15930},{624,33498},{2781,2823},{2979,16260},{3060,30440},{4844,4977},{5959,28213},{6672,32223},{8173,25182},{9437,28229},{14182,25226},{14539,18864},{16268,16462},{25220,25222},{28893,29144}

----------------------------------------------

X(17) --> 

 

 X(15)X(22113) ∩ X(16)X(629) =   

 

= (a^2 - b^2)*(a^2 + b^2 - c^2 + 2*Sqrt[3]*S)*(3*a^2 + 3*b^2 - c^2 + 2*Sqrt[3]*S) + (a^2 - c^2)*(a^2 - b^2 + c^2 + 2*Sqrt[3]*S)*(3*a^2 - b^2 + 3*c^2 + 2*Sqrt[3]*S) : :

 

= lies on these lines: {15,22113},{16,629},{17,622},{30,511},{623,627},{624,6673},{786,4977},{2823,3800},{4802,23066},{5959,7927},{5615,16626},{7684,16629},{9014,29128},{14138,33465},{14369,31939},{14538,22916},{20429,22832},{22737,22907},{22901,33518}

----------------------------------------------

 

X(18) --> 

 

 X(15)X(630) ∩ X(16)X(22114) =   

 

= (a^2 - b^2)*(a^2 + b^2 - c^2 - 2*Sqrt[3]*S)*(3*a^2 + 3*b^2 - c^2 - 2*Sqrt[3]*S) + (a^2 - c^2)*(a^2 - b^2 + c^2 - 2*Sqrt[3]*S)*(3*a^2 - b^2 + 3*c^2 - 2*Sqrt[3]*S) : :

= lies on these lines: {15,630},{16,22114},{18,621},{30,511},{623,6674},{624,628},{786,2781},{4160,28191},{4844,5959},{5611,16627},{7685,16628},{28175,28213},{14139,33464},{14368,31940},{20428,22831},{22736,22861},{22855,33517}

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X(21) --> 

 

 X(11)X(14527) ∩ X(12)X(21) =    

 

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

= lies on these lines: {11,14527},{12,21},{30,511},{36,442},{55,15680},{1319,11281},{2099,14450},{2475,2886},{2823,4977},{2828,4844},{3035,27086},{3419,6763},{3651,30264},{3814,6668},{4996,15326},{5057,10543},{5131,11112},{5176,21677},{5180,6284},{5426,11113},{5427,6667},{5428,31659},{5536,6598},{6175,31157},{7680,13743},{8666,18407},{11263,18990},{11491,11827},{15670,31160},{21155,21161},{22765,26470},{29174,29208},{31254,31260},{31799,32157}

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X(111) --> 

 

 X(1)X(3325) ∩ X(3)X(111) =   

 

= a^2*(a^6*b^2 + a^4*b^4 - a^2*b^6 - b^8 + a^6*c^2 - 14*a^4*b^2*c^2 + 10*a^2*b^4*c^2 + 7*b^6*c^2 + a^4*c^4 + 10*a^2*b^2*c^4 - 20*b^4*c^4 - a^2*c^6 + 7*b^2*c^6 - c^8) : :

= lies on these lines: {1,3325},{3,111},{4,10748},{5,126},{20,14654},{26,14657},{30,511},{49,3048},{99,12093},{140,6719},{143,22100},{182,14688},{351,19901},{352,9871},{376,14666},{381,10717},{382,10734},{549,9172},{574,10204},{4844,7927},{1151,11835},{1152,11836},{1351,10765},{1385,11721},{1482,10704},{1511,9129},{2493,9177},{2823,4802},{5077,7998},{5640,11159},{5650,20326},{5971,13168},{6076,6077},{6093,14514},{6407,11833},{6408,11834},{7514,15560},{9013,28147},{9437,29142},{9734,14662},{10738,10779},{11171,15921},{11287,33879},{12006,14135},{29128,29174},{33532,33900}

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Best regards,
Peter Moses.