HYACINTHOS 28785

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C', A"B"C" the pedal triangles of N = X(5), N* = X(54), resp.

 

Denote:

Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp.
N'a, N'b, N'c = the reflections of Na, Nb, Nc in NA', NB', NC', resp.

N1, N2, N3 = the NPC centers of N*BC, N*CA, N*AB, resp.
N'1, N'2, N'3 = the reflections of N1, N2, N3 in N*A", N*B", N*C", resp.  

Ma, Mb, Mc = the midpoints N'aN'1, N'bN'2, N'cN'3, resp.

 

1. ABC, N'aN'bN'c
2. ABC, N'1N'2N'3
3. ABC, MaMbMc

 

are orthologic.

APH

 

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[Ercole Suppa]

 

Dear Antreas

 

1. The orthology center between ABC and N'aN'bN'c is the point X(1154)

    The orthology center between N'aN'bN'c and ABC is the point X(546)

 

2. The orthology center between ABC and  N'1N'2N'3 is the point X(523)

    The orthology center between N'1N'2N'3 and ABC is the point X(1493)

 

3. The orthology center between ABC and  MaMbMc is the point 

 

W1 = (14 R^2-4 SW)S^6 + (207 R^6-207 R^4 SB-207 R^4 SC-61 R^4 SW+100 R^2 SB SW+100 R^2 SC SW-12 R^2 SW^2-12 SB SW^2-12 SC SW^2+4 SW^3+SB SC (-22 R^2+4 SW))S^4 + (2835 R^10-81 R^8 SB-81 R^8 SC-4365 R^8 SW-18 R^6 SB SW-18 R^6 SC SW+2657 R^6 SW^2+89 R^4 SB SW^2+89 R^4 SC SW^2-813 R^4 SW^3-36 R^2 SB SW^3-36 R^2 SC SW^3+126 R^2 SW^4+4 SB SW^4+4 SC SW^4-8 SW^5+SB SC (99 R^6-91 R^4 SW+36 R^2 SW^2-4 SW^3))S^2 + (-2025 R^10+3465 R^8 SW-2335 R^6 SW^2+773 R^4 SW^3-126 R^2 SW^4+8 SW^5)SB SC: : (barys)

 

= (6-8-13) search numbers [56.8285785223354385, -53.7635900504198008, 14.6330367372740388]

 

 

The orthology center between MaMbMc and ABC is the point

 

W2 = midpoint of X(5) and X(11803)

 

= 2 a^10-13 a^8 b^2+24 a^6 b^4-14 a^4 b^6-2 a^2 b^8+3 b^10-13 a^8 c^2+14 a^6 b^2 c^2+11 a^4 b^4 c^2-3 a^2 b^6 c^2-9 b^8 c^2+24 a^6 c^4+11 a^4 b^2 c^4+10 a^2 b^4 c^4+6 b^6 c^4-14 a^4 c^6-3 a^2 b^2 c^6+6 b^4 c^6-2 a^2 c^8-9 b^2 c^8+3 c^10 : : (barys)

 

= (19 R^2-6 SB-6 SC-4 SW)S^2 + (23 R^2-8 SW) SB SC : : (barys)

 

= X[3]-3*X[8254], 3*X[54]+X[3627], 3*X[195]+5*X[3091], 3*X[1209]-5*X[12812], 3*X[2888]-11*X[5072], X[2914]+X[11801], 7*X[3090]-3*X[21230], 11*X[3525]-3*X[12307], 7*X[3851]+X[11271], X[3853]+X[10619], 7*X[3857]-3*X[6288], 5*X[5076]+3*X[12254], 13*X[5079]+3*X[12316], X[6152]-3*X[13451], 3*X[6689]-2*X[12108], 3*X[7691]-7*X[14869], 3*X[10610]-X[12103], 3*X[12325]-19*X[15022], X[12606]+X[14449], 4*X[12811]-3*X[20584], X[13432]+15*X[19709], X[15704]+3*X[15800]

 

= on lines X(i)X(j) for these {i,j}: {3,8254},{4,17507},{5,1173},{30,12242},{54,3627},{113,137},{143,12010},{195,3091},{539,3850},{1154,3628},{1209,12812},{2888,5072},{2914,11801},{2918,17714},{3090,21230},{3518,15806},{3525,12307},{3851,11271},{3853,10619},{3857,6288},{5076,12254},{5079,12316},{6152,13451},{6689,12108},{7691,14869},{10272,18369},{10610,12103},{12102,18400},{12325,15022},{12606,14449},{12811,20584},{13432,19709},{15425,16337},{15704,15800},{24144,27423}

 

= midpoint of X(i) and X(j) for these {i,j}: {5,11803},{546,1493},{8254,20424},{12606,14449}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {546,22051,1493},{1493,3574,546}

 

= (6-8-13) search numbers [-7.61734693022353381, 4.87972270795838788, 3.77809349803943387]

 

 

Best regards

Ercole Suppa