HYACINTHOS 28615

[Antreas P. Hatzipolakis]:

 

 

 

Let ABC be a triangle and P a point.

Denote:
Pa, Pb, Pc = same to P points of NBC, NCA, NAB, resp.

Which is the locus of P such that the reflections of NPa, NPb, NPc in BC, CA, AB, resp. are concurrent?


[ Angel Montesdeoca]:

 
***  Some cases:

P such that the reflections of NPa, NPb, NPc in BC, CA, AB, resp. are concurrent at  Q.

Pairs {P, Q}:

--- {X(3), X(5944)}
--- {X(4), X(5)}
--- {X(5), X(6150)}
--- {X(6), X(143)}
--- {X(54), X(6343)}
--- {X(64),  Q64}
 
 Q64  = (name pending)

 

4 a^18 -24 a^16 (b^2+c^2) +9 a^14 (7 b^4+10 b^2 c^2+7 c^4) -a^12 (91 b^6+121 b^4 c^2+121 b^2 c^4+91 c^6) +2 a^10 (34 b^8+23 b^6 c^2+24 b^4 c^4+23 b^2 c^6+34 c^8) -a^8 (4 b^10-27 b^8 c^2-13 b^6 c^4-13 b^4 c^6-27 b^2 c^8+4 c^10) -a^6 (b^2-c^2)^2 (41 b^8+61 b^6 c^2+69 b^4 c^4+61 b^2 c^6+41 c^8) +a^4 (b^2-c^2)^2 (37 b^10-14 b^8 c^2-17 b^6 c^4-17 b^4 c^6-14 b^2 c^8+37 c^10) -a^2 (b^2-c^2)^4 (14 b^8-7 b^6 c^2-9 b^4 c^4-7 b^2 c^6+14 c^8) +2 (b^2-c^2)^8 (b^2+c^2) : : 
 
= lies on this line: {1503,3628}

 (6 - 9 - 13) - search numbers  of Q64: (0.522934535696972, -0.673682426567449, 3.86570560690170).


--- {X(74),  Q74}
 
 Q74 = 

 

-12 a^22+93 a^20 (b^2+c^2)-3 (b^2-c^2)^10 (b^2+c^2)-2 a^18 (161 b^4+250 b^2 c^2+161 c^4)+a^16 (649 b^6+1090 b^4 c^2+1090 b^2 c^4+649 c^6)+a^2 (b^2-c^2)^6 (30 b^8-11 b^6 c^2-22 b^4 c^4-11 b^2 c^6+30 c^8)-a^14 (816 b^8+1137 b^6 c^2+1258 b^4 c^4+1137 b^2 c^6+816 c^8)-a^4 (b^2-c^2)^4 (119 b^10-6 b^8 c^2-60 b^6 c^4-60 b^4 c^6-6 b^2 c^8+119 c^10)+a^12 (602 b^10+381 b^8 c^2+412 b^6 c^4+412 b^4 c^6+381 b^2 c^8+602 c^10)+a^10 (-140 b^12+255 b^10 c^2+68 b^8 c^4+66 b^6 c^6+68 b^4 c^8+255 b^2 c^10-140 c^12)+a^6 (b^2-c^2)^2 (236 b^12-3 b^10 c^2-80 b^8 c^4-119 b^6 c^6-80 b^4 c^8-3 b^2 c^10+236 c^12)+a^8 (-198 b^14-25 b^12 c^2+137 b^10 c^4+59 b^8 c^6+59 b^6 c^8+137 b^4 c^10-25 b^2 c^12-198 c^14) : : 
 
 with   (6 - 9 - 13) - search numbers  (0.529704486689447, -0.830678174911267, 3.97127037837397).


--- {X(1173),  Q1173 }
 
 Q1173  =  

= X(4)X(15047)∩X(1263)X(14627)

4 a^16-3 (b^2-c^2)^8-23 a^14 (b^2+c^2)+19 a^2 (b^2-c^2)^6 (b^2+c^2)+a^12 (51 b^4+66 b^2 c^2+51 c^4)-a^10 (47 b^6+43 b^4 c^2+43 b^2 c^4+47 c^6)-a^8 (5 b^8+14 b^6 c^2+6 b^4 c^4+14 b^2 c^6+5 c^8)-a^4 (b^2-c^2)^2 (47 b^8-30 b^6 c^2-45 b^4 c^4-30 b^2 c^6+47 c^8)+3 a^6 (17 b^10-13 b^8 c^2-9 b^6 c^4-9 b^4 c^6-13 b^2 c^8+17 c^10) : : 

 
= lies on these lines:  {4,15047}, {1263,14627}, {5944,13631}, {13163,14706}.
 
 (6 - 9 - 13) - search numbers  of  Q1173: (4.39477987366519, 4.25275934553326, -1.33191346823027).
 
 Angel Montesdeoca