HYACINTHOS 28764

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the pedal triangle of N.

Denote:

Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp.

Oa, Ob, Oc = the circumcenters of NNbNc, NNcNa, NNaNb, resp.

La, Lb, Lc =: NOa, NOb, NOc, resp.

L1, L2, L3 = the reflections of La, Lb, Lc in BC, CA, AB, resp.

 

 

1.The parallels to La, Lb, Lc through A', B', C', resp. are concurrent.  .

 

2. L1, L2, L3 are concurrent.

3. The parallels to L1, L2, L3 through A', B', C', resp. are concurrent.

 

4. the reflections of L1, L2, L3 in NA', NB', NC', resp. are concurrent.

 

 

[Alexandr Skutin]:

 

Found same fact #2.
See it in my blog post 

 

Theorem 7 (part 2) in the comment section  

 

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[Ercole Suppa]

 

Dear Antreas

 

 

*** 1. The parallels to La, Lb, Lc through A', B', C', resp. concur at the point

 

Q1 = X(5)X(128) ∩ X(14071)X(25149)

 

= (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (-a^6+2 a^4 b^2-a^2 b^4-a^2 b^3 c+b^5 c+2 a^4 c^2+a^2 b^2 c^2-a^2 b c^3-2 b^3 c^3-a^2 c^4+b c^5) (a^6-2 a^4 b^2+a^2 b^4-a^2 b^3 c+b^5 c-2 a^4 c^2-a^2 b^2 c^2-a^2 b c^3-2 b^3 c^3+a^2 c^4+b c^5) (-a^10 b^2+4 a^8 b^4-6 a^6 b^6+4 a^4 b^8-a^2 b^10-a^10 c^2+6 a^8 b^2 c^2-9 a^6 b^4 c^2+5 a^4 b^6 c^2-2 a^2 b^8 c^2+b^10 c^2+4 a^8 c^4-9 a^6 b^2 c^4+3 a^2 b^6 c^4-4 b^8 c^4-6 a^6 c^6+5 a^4 b^2 c^6+3 a^2 b^4 c^6+6 b^6 c^6+4 a^4 c^8-2 a^2 b^2 c^8-4 b^4 c^8-a^2 c^10+b^2 c^10) :: (barys)

 

= S^6+S^4 (-9 R^4-5 R^2 SB-5 R^2 SC-3 SB SC+3 R^2 SW+2 SB SW+2 SC SW)+SB SC (18 R^8-6 R^6 SW-7 R^4 SW^2+5 R^2 SW^3-SW^4)+S^2 (114 R^8-40 R^6 SB-40 R^6 SC-126 R^6 SW+46 R^4 SB SW+46 R^4 SC SW+41 R^4 SW^2-17 R^2 SB SW^2-17 R^2 SC SW^2-R^2 SW^3+2 SB SW^3+2 SC SW^3-SW^4+SB SC (-53 R^4+33 R^2 SW-4 SW^2)) : : (barys)

 

= lies on these lines: {5,128}, {14071,25149}

 

= (6-8-13) search numbers [-0.362371496744293892, -0.860562214879911408, 4.40368670609091495]

 

 

*** 2. L1, L2, L3 concur at the point 

 

Q2 = X(5)X(51) ∩ X(110)X(1157)

 

= a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^14 b^2-6 a^12 b^4+15 a^10 b^6-20 a^8 b^8+15 a^6 b^10-6 a^4 b^12+a^2 b^14+a^14 c^2-9 a^12 b^2 c^2+25 a^10 b^4 c^2-30 a^8 b^6 c^2+15 a^6 b^8 c^2-a^4 b^10 c^2-a^2 b^12 c^2-6 a^12 c^4+25 a^10 b^2 c^4-30 a^8 b^4 c^4+9 a^6 b^6 c^4+3 a^4 b^8 c^4-2 a^2 b^10 c^4+b^12 c^4+15 a^10 c^6-30 a^8 b^2 c^6+9 a^6 b^4 c^6-a^4 b^6 c^6+2 a^2 b^8 c^6-4 b^10 c^6-20 a^8 c^8+15 a^6 b^2 c^8+3 a^4 b^4 c^8+2 a^2 b^6 c^8+6 b^8 c^8+15 a^6 c^10-a^4 b^2 c^10-2 a^2 b^4 c^10-4 b^6 c^10-6 a^4 c^12-a^2 b^2 c^12+b^4 c^12+a^2 c^14) :: (barys)

 

= S^4 (R^2-SB-SC+SW)+SB SC (15 R^6-18 R^4 SW+7 R^2 SW^2-SW^3)+S^2 (-17 R^6-8 R^4 SB-8 R^4 SC+SB SC (5 R^2-SW)+22 R^4 SW+6 R^2 SB SW+6 R^2 SC SW-9 R^2 SW^2-SB SW^2-SC SW^2+SW^3) : : (barys)

 

= lies on these lines: {5,51}, {110,1157}, {5944,6150}

 

= (6-8-13) search numbers [1.09276477109133165, -0.520384412072055508, 3.49657764206903369]

 

 

*** 3. The parallels to L1, L2, L3 through A', B', C', resp. concur at the point

 

Q3 = X(5)X(51) ∩ X(1510)X(6150)

 

= a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-b^2 c^2+c^4) (a^10 b^2-4 a^8 b^4+6 a^6 b^6-4 a^4 b^8+a^2 b^10+a^10 c^2-6 a^8 b^2 c^2+9 a^6 b^4 c^2-5 a^4 b^6 c^2+2 a^2 b^8 c^2-b^10 c^2-4 a^8 c^4+9 a^6 b^2 c^4-3 a^2 b^6 c^4+4 b^8 c^4+6 a^6 c^6-5 a^4 b^2 c^6-3 a^2 b^4 c^6-6 b^6 c^6-4 a^4 c^8+2 a^2 b^2 c^8+4 b^4 c^8+a^2 c^10-b^2 c^10) :: (barys)

 

= S^4 (3 R^2-SB-SC+SW)+SB SC (6 R^6-10 R^4 SW+5 R^2 SW^2-SW^3)+S^2 (-26 R^6-8 R^4 SB-8 R^4 SC+SB SC (7 R^2-SW)+30 R^4 SW+6 R^2 SB SW+6 R^2 SC SW-11 R^2 SW^2-SB SW^2-SC SW^2+SW^3) : : (barys)

 

= lies on these lines: {5,51}, {1510,6150}, {12060,18350}

 

= (6-8-13) search numbers [0.822818256363118121, -0.419245541474519416, 3.55114912306837649]

 

 

*** 4. the reflections of L1, L2, L3 in NA', NB', NC', resp. concur at the point 

 

Q4 = X(5)X(128) ∩ X(6343)X(25149)

 

= (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (a^22 b^2-8 a^20 b^4+28 a^18 b^6-56 a^16 b^8+70 a^14 b^10-56 a^12 b^12+28 a^10 b^14-8 a^8 b^16+a^6 b^18+a^22 c^2-13 a^20 b^2 c^2+56 a^18 b^4 c^2-119 a^16 b^6 c^2+140 a^14 b^8 c^2-91 a^12 b^10 c^2+28 a^10 b^12 c^2-a^8 b^14 c^2-a^6 b^16 c^2-8 a^20 c^4+56 a^18 b^2 c^4-141 a^16 b^4 c^4+163 a^14 b^6 c^4-79 a^12 b^8 c^4-7 a^10 b^10 c^4+30 a^8 b^12 c^4-23 a^6 b^14 c^4+13 a^4 b^16 c^4-5 a^2 b^18 c^4+b^20 c^4+28 a^18 c^6-119 a^16 b^2 c^6+163 a^14 b^4 c^6-79 a^12 b^6 c^6+5 a^10 b^8 c^6-11 a^8 b^10 c^6+35 a^6 b^12 c^6-39 a^4 b^14 c^6+25 a^2 b^16 c^6-8 b^18 c^6-56 a^16 c^8+140 a^14 b^2 c^8-79 a^12 b^4 c^8+5 a^10 b^6 c^8+7 a^8 b^8 c^8-12 a^6 b^10 c^8+39 a^4 b^12 c^8-45 a^2 b^14 c^8+28 b^16 c^8+70 a^14 c^10-91 a^12 b^2 c^10-7 a^10 b^4 c^10-11 a^8 b^6 c^10-12 a^6 b^8 c^10-26 a^4 b^10 c^10+25 a^2 b^12 c^10-56 b^14 c^10-56 a^12 c^12+28 a^10 b^2 c^12+30 a^8 b^4 c^12+35 a^6 b^6 c^12+39 a^4 b^8 c^12+25 a^2 b^10 c^12+70 b^12 c^12+28 a^10 c^14-a^8 b^2 c^14-23 a^6 b^4 c^14-39 a^4 b^6 c^14-45 a^2 b^8 c^14-56 b^10 c^14-8 a^8 c^16-a^6 b^2 c^16+13 a^4 b^4 c^16+25 a^2 b^6 c^16+28 b^8 c^16+a^6 c^18-5 a^2 b^4 c^18-8 b^6 c^18+b^4 c^20) :: (barys)

 

= S^6+S^4 (R^4-5 R^2 SB-5 R^2 SC-3 SB SC-R^2 SW+2 SB SW+2 SC SW)+SB SC (-27 R^8+52 R^6 SW-33 R^4 SW^2+9 R^2 SW^3-SW^4)+S^2 (69 R^8-40 R^6 SB-40 R^6 SC-68 R^6 SW+46 R^4 SB SW+46 R^4 SC SW+15 R^4 SW^2-17 R^2 SB SW^2-17 R^2 SC SW^2+3 R^2 SW^3+2 SB SW^3+2 SC SW^3-SW^4+SB SC (-43 R^4+29 R^2 SW-4 SW^2)) : : (barys)

 

= lies on these lines: {5,128}, {6343,25149}

 

= (6-8-13) search numbers [-1.27761473512349238, -1.40301775888283949, 5.20165280811411062]

 

 

Best regards

Ercole Suppa