HYACINTHOS 26047

[Antreas P. Hatzipolakis]:

 

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

Denote:

A"B"C" = the reflection triangle
(ie A", B", C" = the reflections of A,B,C in BC, CA, AB, resp.)

Ab, Ac = the orthogonal projections of A" on BB', CC', resp.

Na = the NPC center of A"AbAc. Similarly Nb, Nc.

1. ABC, NaNbNc are orthologic.

2. A'B'C', NaNbNc are orthologic
3. A"B"C", NaNbNc are orthologic. 

The orthologic center (NaNbNc, A"B"C") is the orthocenter of A"B"C"

 

[Peter Moses]:

 

 

Hi Antreas,

 

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

1a). ABC NaNbNc:

(a^8-4 a^6 b^2+6 a^4 b^4-4 a^2 b^6+b^8-2 a^6 c^2+3 a^4 b^2 c^2+3 a^2 b^4 c^2-2 b^6 c^2-a^2 b^2 c^4+2 a^2 c^6+2 b^2 c^6-c^8) (a^8-2 a^6 b^2+2 a^2 b^6-b^8-4 a^6 c^2+3 a^4 b^2 c^2-a^2 b^4 c^2+2 b^6 c^2+6 a^4 c^4+3 a^2 b^2 c^4-4 a^2 c^6-2 b^2 c^6+c^8):: 

on lines {{3,12325},{74,12254},{265, 2888},{1154,3521},{1176,5965}, {9706,11271}}.

on Jerabek.

isoconjugate of X(2216) and X(6153).

barycentric quotient X(570) / X(6153).

Searches: {-742. 392664497152243943454954262, 611. 441574892485705967838392750,- 77. 0222729834354585930213217572}.

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

 

1b). NaNbNc ABC:

2 a^10-4 a^8 b^2+a^6 b^4+a^4 b^6+a^2 b^8-b^10-4 a^8 c^2-a^4 b^4 c^2+2 a^2 b^6 c^2+3 b^8 c^2+a^6 c^4-a^4 b^2 c^4-6 a^2 b^4 c^4-2 b^6 c^4+a^4 c^6+2 a^2 b^2 c^6-2 b^4 c^6+a^2 c^8+3 b^2 c^8-c^10:: 

on lines {{3,2916},{4,54},{24,11550},{ 30,1216},{52,542},{125,3518},{ 143,10116},{182,7528},{185, 7576},{211,2794},{235,7687},{ 381,11750},{389,1503},{403, 11572},{427,10282},{428,6146}, {511,7553},{539,10263},{1092, 7391},{1112,10114},{1209,2937} ,{1495,1594},{1539,3627},{ 1595,11430},{1598,1619},{1853, 3517},{2777,6240},{3426,5925}, {3541,11202},{3543,12278},{ 3575,6000},{3853,12897},{5092, 7405},{5446,10112},{5899,6288} ,{5900,12244},{5965,6243},{ 5972,13371},{6145,10117},{ 6242,7731},{6800,7566},{7487, 11438},{7530,9927},{7544, 10984},{9729,11645},{13163, 13363}}.

midpoint of X(i) and X(j) for these {i,j}: {{6240, 11381}, {7553, 12134}}.

reflection of X(i) in X(j) for these {i,j}: {{389, 6756}, {6146, 10110}, {10112, 5446}, {10114, 1112}, {10116, 143}, {12897, 3853}, {13403, 4}}.

3 X[428] - X[6146], X[52] - 3 X[7540], X[185] - 3 X[7576], 3 X[428] - 2 X[10110], 4 X[143] - 3 X[11225], 2 X[10116] - 3 X[11225], 3 X[389] - 4 X[11745], 3 X[6756] - 2 X[11745], 3 X[381] - X[11750], 3 X[3543] + X[12278], 5 X[4] - X[12289], 4 X[13163] - 3 X[13363], 2 X[12289] - 5 X[13403].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1614,3574),(4,1629,6750),( 4,9833,578),(143,10116,11225), (428,6146,10110).

crosspoint of X(4) and X(11816).

crosssum of X(3) and X(6101).

Searches: {-5. 65106782232459383390229978998, -7. 39525863228642826056813410806, 11. 3686440684094112068512024650}.

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

 

2a). A'B'C' NaNbNc:

(a^2+b^2-c^2) (a^2-b^2+c^2) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2+3 a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (3 a^6-7 a^4 b^2+5 a^2 b^4-b^6-7 a^4 c^2-3 a^2 b^2 c^2+b^4 c^2+5 a^2 c^4+b^2 c^4-c^6):: 

on lines {{4,12175},{52,11271},{185, 12254},{1986,6242},{3518,5898} }.

on Feuerbach of the orthic triangle.

X(4)-Ceva conjugate of X(6143).

Searches: {1098. 37516893532765910533657879,- 898. 919202674353038679900680742, 119. 026957594000640140586560020}

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

 

2b). NaNbNc A'B'C'

a^2 (3 a^6 b^2-9 a^4 b^4+9 a^2 b^6-3 b^8+3 a^6 c^2-10 a^4 b^2 c^2+7 b^6 c^2-9 a^4 c^4-8 b^4 c^4+9 a^2 c^6+7 b^2 c^6-3 c^8):: 

on lines {{4,93},{26,11477},{30,10112}, {52,550},{140,143},{546,12002} ,{568,3522},{576,7525},{1176, 1351},{1216,13364},{1493,2937} ,{1656,3060},{1657,6102},{ 3523,5946},{3532,12084},{3850, 5446},{3851,11412},{3858,5562} ,{5073,5889},{10299,13340},{ 10625,12006}}.

midpoint of X(6243) and X(10263).

reflection of X(i) in X(j) for these {i,j}: {{6101, 10095}, {10625, 12006}, {10627, 143}, {11591, 5446}}.

2 X[140] - 3 X[143], 3 X[52] - X[550], 5 X[1656] - 9 X[3060], 9 X[568] - 5 X[3522], 2 X[3850] - 3 X[5446], 7 X[140] - 6 X[5447], 7 X[143] - 4 X[5447], 5 X[5447] - 7 X[5462], 5 X[140] - 6 X[5462], 5 X[143] - 4 X[5462], 5 X[3858] - 3 X[5562], 5 X[4] - 3 X[5876], X[5073] + 3 X[5889], 7 X[3523] - 9 X[5946], 5 X[1656] - 3 X[6101], 3 X[3060] - X[6101], X[1657] - 3 X[6102], X[4] + 3 X[6243], X[5876] + 5 X[6243], 15 X[3060] - 7 X[7999], 5 X[6101] - 7 X[7999], 7 X[7999] - 10 X[10095], 5 X[1656] - 6 X[10095], 3 X[3060] - 2 X[10095], X[5876] - 5 X[10263], X[4] - 3 X[10263], 8 X[5447] - 7 X[10627], 8 X[5462] - 5 X[10627], 4 X[140] - 3 X[10627], 7 X[3851] - 3 X[11412], 4 X[3850] - 3 X[11591], 11 X[10627] - 16 X[11695], 11 X[5447] - 14 X[11695], 11 X[140] - 12 X[11695], 11 X[5462] - 10 X[11695], 11 X[143] - 8 X[11695], 3 X[546] - 4 X[12002], 13 X[10299] - 9 X[13340], 16 X[5462] - 15 X[13363], 8 X[140] - 9 X[13363], 4 X[143] - 3 X[13363], 2 X[10627] - 3 X[13363], 2 X[1216] - 3 X[13364].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (143,10627,13363),(3060,6101, 10095).

Searches: {-7. 62172712609987426080912723998, -5. 80240510039145235863993404587, 11. 1754343788399419333194458604}.

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

 

3a). A''B''C'' NaNbNc:

7 a^16-38 a^14 b^2+78 a^12 b^4-62 a^10 b^6-20 a^8 b^8+78 a^6 b^10-62 a^4 b^12+22 a^2 b^14-3 b^16-38 a^14 c^2+134 a^12 b^2 c^2-160 a^10 b^4 c^2+84 a^8 b^6 c^2-74 a^6 b^8 c^2+106 a^4 b^10 c^2-64 a^2 b^12 c^2+12 b^14 c^2+78 a^12 c^4-160 a^10 b^2 c^4+79 a^8 b^4 c^4-4 a^6 b^6 c^4-41 a^4 b^8 c^4+60 a^2 b^10 c^4-12 b^12 c^4-62 a^10 c^6+84 a^8 b^2 c^6-4 a^6 b^4 c^6-6 a^4 b^6 c^6-18 a^2 b^8 c^6-12 b^10 c^6-20 a^8 c^8-74 a^6 b^2 c^8-41 a^4 b^4 c^8-18 a^2 b^6 c^8+30 b^8 c^8+78 a^6 c^10+106 a^4 b^2 c^10+60 a^2 b^4 c^10-12 b^6 c^10-62 a^4 c^12-64 a^2 b^2 c^12-12 b^4 c^12+22 a^2 c^14+12 b^2 c^14-3 c^16:: 

on line {11558,12316}.

Searches: {2939. 14300236780756215412811185,- 2409. 27998024119178332763975423, 315. 076188171436738874194441796}.

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

 

3b). NaNbNc A''B''C'':

a^2 (a^12 b^2-4 a^10 b^4+5 a^8 b^6-5 a^4 b^10+4 a^2 b^12-b^14+a^12 c^2-7 a^10 b^2 c^2+11 a^8 b^4 c^2-5 a^6 b^6 c^2+4 a^4 b^8 c^2-8 a^2 b^10 c^2+4 b^12 c^2-4 a^10 c^4+11 a^8 b^2 c^4-5 a^6 b^4 c^4+a^4 b^6 c^4+3 a^2 b^8 c^4-6 b^10 c^4+5 a^8 c^6-5 a^6 b^2 c^6+a^4 b^4 c^6+2 a^2 b^6 c^6+3 b^8 c^6+4 a^4 b^2 c^8+3 a^2 b^4 c^8+3 b^6 c^8-5 a^4 c^10-8 a^2 b^2 c^10-6 b^4 c^10+4 a^2 c^12+4 b^2 c^12-c^14):: 

on lines {{2,6153},{3,13368},{6,24},{ 22,10203},{143,9706},{156,195} ,{382,1154},{399,10263},{511, 12325},{1209,7999},{1614,9920} ,{2888,11412},{5640,8254},{ 5890,12254},{6241,6242},{6288, 11459},{6689,11465},{7512, 11649},{7691,12084},{9512, 11816},{9781,11808},{10628, 12290},{10938,12289},{11061, 11271},{11451,13365}}.

reflection of X(i) in X(j) for these {i,j}: {{3, 13368}, {54, 6152}, {11412, 2888}, {12226, 1209}, {12291, 54}, {12316, 10263}}.

3 X[54] - 4 X[973], 2 X[195] - 3 X[3060], 16 X[973] - 15 X[3567], 4 X[54] - 5 X[3567], 5 X[3567] - 8 X[6152], 2 X[973] - 3 X[6152], 3 X[2] - 4 X[6153], X[6241] - 4 X[6242], 8 X[973] - 9 X[7730], 5 X[3567] - 6 X[7730], 2 X[54] - 3 X[7730], 4 X[6152] - 3 X[7730], 8 X[1209] - 7 X[7999], 9 X[5640] - 8 X[8254], 4 X[6288] - 3 X[11459], 16 X[6689] - 17 X[11465], 15 X[7730] - 8 X[11577], 5 X[54] - 4 X[11577], 5 X[973] - 3 X[11577], 5 X[6152] - 2 X[11577], 7 X[9781] - 8 X[11808], 7 X[7999] - 4 X[12226], 3 X[5890] - 2 X[12254], 8 X[11577] - 5 X[12291], 8 X[973] - 3 X[12291], 5 X[3567] - 2 X[12291], 4 X[6152] - X[12291], 3 X[7730] - X[12291], 15 X[11451] - 16 X[13365].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54,2917,11464),(54,6152,7730) ,(54,7730,3567),(54,12380, 2917),(6152,12291,3567),(7730, 12291,54).

Searches: {31. 1514091170463906328131471682,- 37. 4232804632210505941401421160, 15. 1715159793467722993016171052}.

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

 

Best regards,

Peter Moses.