HYACINTHOS 29448

[Antreas P. Hatzipolakis]:

Let ABc be a triangle, A'B'C' the pedal triangle of I and IaIbIc the antipedal triangle of I.

Denote:

La, Lb, Lc = the Euler lines of IBC, ICA, IAB, resp. (concurrent at X(21))

A* = (reflection of Lb in AI) /\ (reflection of Lc in AI)
B* = (reflection of Lc in BI) /\ (reflection of La in BI)
C* = (reflection of La in CI) /\ (reflection of Lb in CI)

 

 

1. ABC, A*B*C* are perspective at the orthocenter of A'B'C'

 

2. A'B'C', A*B*C* are circumorthologic.  
The orthologic center U = (A'B'C', A*B*C*) lies on the circumcircle of A'B'C' (incircle)

The orthologic center W = (A*B*C*, A'B'C') lies on the cicumcircle of A*B*C* which is the circle (I, IX(21))

3. A'B'C', A*B*C* are parallelogic.
The parallelogic center U' = (A'B'C', A*B*C*) lies circumcircle of A'B'C' (incircle) 

The parallelogic center W' = (A*B*C*, A'B'C')  lies on the cicumcircle of A*B*C* and is X(21)

 

U, U' are antipodes in the circumcircle of A'B'C' (incircle)
W, W' are antipodes in the circumcircle of A*B*C*.

Also A*B*C*, IaIbIc are circumorthologic and circumparallelogic.
And since A'B'C', IaIbIc are homothetic, the orthologic center (A*B*C*, IaIbIc) = W and the parallelogic center (A*B*C*, IaIbIc) = W' = X(21)

Orthologic center (IaIbIc, A*B*C*) = U", Parallelogic center (IaIbIc, A*B*C*) = W", antipodes in the circumcircle of IaIbIc

Which are the points U, W, U', U", W" ?

 

 

[Peter Moses]:


Hi Antreas,

1).
X(65).

2).
U  = REFLECTION OF X(1365) IN X(1) =

 

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

= lies on the incircle and these lines: {1,1365}, {11,214}, {36,14027}, {55,759}, {56,6011}, {942,1357}, {1283,8240}, {1356,21746}, {1358,3664}, {1364,10544}, {1682,6044}, {1697,21381}, {3022,11997}, {3024,3057},  {3295,14663}, {3323,5088},{3326,5497}, {3685,4542}, {4092,6740}, {4313,19642}, {5048,31522}, {5441,12896}

= reflection of X(1365) in X(1)


W =  REFLECTION OF X(21) IN X(1) =

 

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

= 3 X[1] - X[191], 5 X[1] - 2 X[3647], 5 X[1] - 3 X[5426], 4 X[1] - X[11684], 3 X[2] - 4 X[11281], X[8] - 4 X[16137], 2 X[10] - 3 X[26725], 4 X[10] - 5 X[31254], 3 X[21] - 2 X[191], 5 X[21] - 4 X[3647], 5 X[21] - 6 X[5426], X[21] + 2 X[16126], X[79] + 2 X[3244], X[145] + 2 X[3649], 5 X[191] - 6 X[3647], 5 X[191] - 9 X[5426], 4 X[191] - 3 X[11684], X[191] + 3 X[16126], 3 X[354] - 2 X[8261], 4 X[551] - 3 X[15671], 4 X[1385] - 3 X[21161], 3 X[3241] + X[14450], 2 X[3243] + X[16133], 5 X[3616] - 4 X[6675], 7 X[3622] - 5 X[15674], 7 X[3622] - 4 X[18253], 5 X[3623] - 2 X[10543], 5 X[3623] - X[15680], X[3632] - 4 X[6701], 4 X[3635] - X[5441], 2 X[3647] - 3 X[5426], 8 X[3647] - 5 X[11684], 2 X[3647] + 5 X[16126], X[3648] - 4 X[15174], X[3648] - 7 X[20057], 2 X[3652] - 3 X[28461], X[3652] - 4 X[33179], 12 X[5426] - 5 X[11684], 3 X[5426] + 5 X[16126], 2 X[5428] - 3 X[10246], 3 X[5603] - 2 X[6841], 3 X[6175] - 4 X[11263], 2 X[7982] + X[33557], 2 X[10021] - 3 X[10283], 3 X[10032] - 2 X[31888], 6 X[10222] - X[16138], 4 X[10222] - X[21669], 3 X[10247] - X[13743], 3 X[11224] + X[16143], X[11684] + 4 X[16126], X[12531] - 4 X[33593], 4 X[13607] - X[16113], 4 X[15174] - 7 X[20057], 5 X[15674] - 4 X[18253], 3 X[15677] - X[31888], 2 X[16138] - 3 X[21669], 2 X[16139] - 3 X[21161], 2 X[19919] - 3 X[28453], 6 X[26725] - 5 X[31254], 3 X[28461] - 8 X[33179]

= lies on these lines: {1,21}, {2,11281}, {3,31660}, {8,442}, {10,5425}, {30,944}, {35,4084}, {56,27086}, {65,100}, {72,5260}, {78,11529}, {79,1320}, {104,24475}, {145,388}, {214,3337}, {226,5086}, {354,8261}, {355,33592}, {390,2098}, {404,5902}, {484,4757}, {517,3651}, {518,15988}, {519,5178}, {551,15671}, {644,3970}, {908,6738}, {938,26129}, {942,4511}, {946,10707}, {950,5057}, {952,1389}, {960,5284}, {1043,17164}, {1125,4867}, {1159,5687}, {1201,3315}, {1210,31272}, {1280,1432}, {1317,12913}, {1385,16139}, {1388,5427}, {1392,10308}, {1434,17136}, {1466,18419}, {1476,5083}, {1770,11015}, {1837,31053}, {1844,17519}, {1999,10474}, {2136,3340}, {2287,2294}, {2550,20013}, {2646,3218}, {2771,7984}, {2795,7983}, {3057,3957}, {3219,3962}, {3242,28369}, {3243,16133}, {3336,13587}, {3339,4855}, {3485,11680}, {3486,5905}, {3488,11415}, {3555,4861}, {3616,5730}, {3622,5289}, {3632,6701}, {3635,5441}, {3648,15174}, {3652,28461}, {3671,20292}, {3681,11523}, {3689,10107}, {3753,4420}, {3812,9342}, {3871,5903}, {3872,8000}, {3885,25415}, {3924,32911}, {3935,5836},{3940,9780}, {4018,24929}, {4067,5251}, {4134,32635}, {4188,5221}, {4360,17220}, {4430,12513}, {4463,17016}, {4647,4720}, {4666,15829}, {4673,20929}, {4880,5267}, {4881,32636}, {4930,15670}, {5047,5692}, {5048,17637}, {5180,15171}, {5204,23958}, {5249,6737}, {5428,10246}, {5440,31794}, {5499,5844}, {5506,17547}, {5554,25568}, {5603,6841}, {5691,31164}, {5693,6912}, {5694,6920}, {5794,31019},  {5835,33175}, {5855,15888}, {5883,17531}, {5884,6909}, {5885,6940}, {5904,30147}, {6224,18990}, {6265,6583}, {6326,6915}, {6831,9803}, {6986,31806}, {7354,17483}, {7966,7982}, {7971,10864}, {7990,11224}, {8148,16117}, {8422,16147}, {9528,10701}, {9963,11552}, {10021,10283}, {10032,15677}, {10057,12531}, {10176,17536}, {10247,13743}, {10523,11681}, {10572,16155}, {10595,11240}, {10609,24470}, {10890,16124}, {10950,20060}, {11239,12245}, {11518,19861}, {11551,17647}, {12560,25722}, {12653,13146}, {12709,16465}, {13465,31649}, {13607,16113}, {14110,18444}, {14563,21075}, {15178,22937}, {16141,33176}, {17541,30139}, {17605,20288}, {18398,30144}, {19919,28453}, {20040,32922}, {20586,33667}, {21285,33949}, {21740,24474}, {23345,28217}, {23536,26729}, {24391,24541}, {27714,31247}, {28212,31651}, {31663,33595}

= midpoint of X(i) and X(j) for these {i,j}: {1, 16126}, {145, 2475}, {7982, 16132}, {8148, 16117}, {12653, 13146}}.
= reflection of X(i) in X(j) for these {i,j}: {{8, 442}, {21, 1}, {355, 33592}, {442, 16137}, {2475, 3649}, {3651, 33858}, {10032, 15677}, {11684, 21}, {13465, 31649}, {15680, 10543}, {16139, 1385}, {21677, 11281}, {22937, 15178}, {33557, 16132}

= anticomplement of X(21677).
= X(17097)-anticomplementary conjugate of X(1330)
= barycentric product X(81)*X(27690)
= barycentric quotient X (27690)/X(321)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2650, 81}, {1, 3868, 2975}, {1, 3869, 1621}, {1, 3894, 8666}, {1, 3901, 993}, {1, 11520, 3873}, {1, 11533, 1962}, {1, 11682, 3890}, {1, 12559, 3868}, {10, 26725, 31254}, {942, 4511, 5253}, {1385, 16139, 21161}, {2646, 3218, 5303}, {3244, 11009, 1320}, {3339, 4855, 9352}, {3340, 3870, 14923}, {3485, 12649, 11680}, {3647, 5426, 21}, {3648, 20057, 15174}, {5692, 30143, 5047}, {5730, 15934, 3616}, {5902, 22836, 404}, {6326, 31870, 6915}, {6737, 12563, 5249}, {11281, 21677, 2}, {11523, 19860, 3681}

3).
U' = X(1365).

W' = X(21).

4).
U" = REFLECTION OF X(1) IN X(6011) =

 

= a*(a^9 - 3*a^7*b^2 + a^6*b^3 + 3*a^5*b^4 - 3*a^4*b^5 - a^3*b^6 + 3*a^2*b^7 - b^9 - 8*a^7*b*c + 7*a^6*b^2*c + 8*a^5*b^3*c - 14*a^4*b^4*c + 4*a^3*b^5*c + 3*a^2*b^6*c - 4*a*b^7*c + 4*b^8*c - 3*a^7*c^2 + 7*a^6*b*c^2 + 3*a^5*b^2*c^2 + a^4*b^3*c^2 - a^3*b^4*c^2 - 7*a^2*b^5*c^2 + a*b^6*c^2 - b^7*c^2 + a^6*c^3 + 8*a^5*b*c^3 + a^4*b^2*c^3 - 16*a^3*b^3*c^3 + 9*a^2*b^4*c^3 + 4*a*b^5*c^3 - 11*b^6*c^3 + 3*a^5*c^4 - 14*a^4*b*c^4 - a^3*b^2*c^4 + 9*a^2*b^3*c^4 - 2*a*b^4*c^4 + 9*b^5*c^4 - 3*a^4*c^5 + 4*a^3*b*c^5 - 7*a^2*b^2*c^5 + 4*a*b^3*c^5 + 9*b^4*c^5 - a^3*c^6 + 3*a^2*b*c^6 + a*b^2*c^6 - 11*b^3*c^6 + 3*a^2*c^7 - 4*a*b*c^7 - b^2*c^7 + 4*b*c^8 - c^9) : :
= 3 X[165] - 2 X[759], 3 X[1699] - 4 X[31845]

= lies on the Bevan circle and these lines: {1,6011}, {3,1054}, {20,1768}, {40,21381}, {165,759}, {573,5540},{1365,1697}, {1695,6044}, {1699,31845}, {1742,5539}, {3579,14663}, {5535,16528}, {7991,9904}

= reflection of X(i) in X(j) for these {i,j}: {1, 6011}, {14663, 3579}, {21381, 40}
= excentral isogonal conjugate of X (758)


W" = X(21381).

Best regards,
Peter Moses.