HYACINTHOS 28869

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A1B1C1 the pedal triangle of I..

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

(Nab), (Nac) = the NPCs of NaIB, NaIC, resp.
Ra = the radical axis of (Nab), (Nac)
Similarly Rb, Rc

A*B*C* = the triangle bounded by Ra, Rb, Rc.

1. ABC, A*B*C* are perspective.

 

2.  A1B1C1, A*B*C* are orthologic

3.  A1B1C1, A*B*C* are parallelogic.

 

 

[Peter Moses]:

Hi Antreas,

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Perspective ABC, A*B*C*; 

 

= a*(2*a^9 - 3*a^8*b - 4*a^7*b^2 + 7*a^6*b^3 - 3*a^4*b^5 + 4*a^3*b^6 - 3*a^2*b^7 - 2*a*b^8 + 2*b^9 - a^8*c + 5*a^7*b*c - a^6*b^2*c - 5*a^5*b^3*c + 4*a^4*b^4*c - 4*a^3*b^5*c + 4*a*b^7*c - 2*b^8*c - 3*a^7*c^2 - a^6*b*c^2 + 9*a^5*b^2*c^2 - 8*a^4*b^3*c^2 - 6*a^3*b^4*c^2 + 12*a^2*b^5*c^2 - 3*b^7*c^2 + 3*a^6*c^3 - 5*a^5*b*c^3 - 4*a^4*b^2*c^3 + 14*a^3*b^3*c^3 - 6*a^2*b^4*c^3 - 4*a*b^5*c^3 + 4*b^6*c^3 - a^5*c^4 + 8*a^4*b*c^4 - 4*a^3*b^2*c^4 - 8*a^2*b^3*c^4 + 4*a*b^4*c^4 - 3*b^5*c^4 - a^4*c^5 - 5*a^3*b*c^5 + 9*a^2*b^2*c^5 - 5*a*b^3*c^5 + 3*a^3*c^6 - a^2*b*c^6 - a*b^2*c^6 + 7*b^3*c^6 - 3*a^2*c^7 + 5*a*b*c^7 - 4*b^2*c^7 - a*c^8 - 3*b*c^8 + 2*c^9)*(2*a^9 - a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4 - a^4*b^5 + 3*a^3*b^6 - 3*a^2*b^7 - a*b^8 + 2*b^9 - 3*a^8*c + 5*a^7*b*c - a^6*b^2*c - 5*a^5*b^3*c + 8*a^4*b^4*c - 5*a^3*b^5*c - a^2*b^6*c + 5*a*b^7*c - 3*b^8*c - 4*a^7*c^2 - a^6*b*c^2 + 9*a^5*b^2*c^2 - 4*a^4*b^3*c^2 - 4*a^3*b^4*c^2 + 9*a^2*b^5*c^2 - a*b^6*c^2 - 4*b^7*c^2 + 7*a^6*c^3 - 5*a^5*b*c^3 - 8*a^4*b^2*c^3 + 14*a^3*b^3*c^3 - 8*a^2*b^4*c^3 - 5*a*b^5*c^3 + 7*b^6*c^3 + 4*a^4*b*c^4 - 6*a^3*b^2*c^4 - 6*a^2*b^3*c^4 + 4*a*b^4*c^4 - 3*a^4*c^5 - 4*a^3*b*c^5 + 12*a^2*b^2*c^5 - 4*a*b^3*c^5 - 3*b^4*c^5 + 4*a^3*c^6 + 4*b^3*c^6 - 3*a^2*c^7 + 4*a*b*c^7 - 3*b^2*c^7 - 2*a*c^8 - 2*b*c^8 + 2*c^9) : : 

 

= lies on these lines: {}

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Orthologic A1B1C1, A*B*C*; 

= X(11)X(523)∩X(30)X(1317)

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

 

= lies on the incircle and these lines: {11, 523}, {30, 1317}, {55, 1290}, {56, 2687}, {513, 3024}, {517, 3028}, {1319, 1354}, {1360, 2078}, {3021, 5160}, {3319, 11011}, {4854, 6023}, {9957, 13756}

= reflection of X(3024) in the X(1)X(3) line

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Orthologic A*B*C*, A1B1C1; 

= X(1)X(8674)∩X(104)X(7978)

= a*(2*a^9 - 2*a^8*b - 5*a^7*b^2 + 4*a^6*b^3 + 3*a^5*b^4 + a^3*b^6 - 4*a^2*b^7 - a*b^8 + 2*b^9 - 2*a^8*c + 8*a^7*b*c - 8*a^5*b^3*c + 4*a^4*b^4*c - 7*a^3*b^5*c + a^2*b^6*c + 7*a*b^7*c - 3*b^8*c - 5*a^7*c^2 + 8*a^5*b^2*c^2 - 6*a^4*b^3*c^2 - 4*a^3*b^4*c^2 + 11*a^2*b^5*c^2 + a*b^6*c^2 - 5*b^7*c^2 + 4*a^6*c^3 - 8*a^5*b*c^3 - 6*a^4*b^2*c^3 + 22*a^3*b^3*c^3 - 8*a^2*b^4*c^3 - 7*a*b^5*c^3 + 5*b^6*c^3 + 3*a^5*c^4 + 4*a^4*b*c^4 - 4*a^3*b^2*c^4 - 8*a^2*b^3*c^4 + b^5*c^4 - 7*a^3*b*c^5 + 11*a^2*b^2*c^5 - 7*a*b^3*c^5 + b^4*c^5 + a^3*c^6 + a^2*b*c^6 + a*b^2*c^6 + 5*b^3*c^6 - 4*a^2*c^7 + 7*a*b*c^7 - 5*b^2*c^7 - a*c^8 - 3*b*c^8 + 2*c^9) : : 

 

= X[13211] - 3 X[16173]

 

= lies on these lines: {1, 8674}, {104, 7978}, {110, 1320}, {113, 952}, {119, 11723}, {125, 1387}, {944, 10767}, {1145, 5972}, {2771, 7984}, {2776, 13868}, {2802, 11720}, {10738, 12898}, {13211, 16173}

= midpoint of X(i) and X(j) for these {i,j}: {104, 7978}, {110, 1320}, {944, 10767}, {10738, 12898}
= reflection of X(i) in X(j) for these {i,j}: {119, 11723}, {125, 1387}, {1145, 5972}

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Parallelogic A1B1C1, A*B*C*; 

= X(11)X(30)∩X(12)X(2222)

= (a + b - c)*(a - b + c)*(2*a^4 - a^3*b - a^2*b^2 + a*b^3 - b^4 - a^3*c - a^2*c^2 + 2*b^2*c^2 + a*c^3 - c^4)^2 : : 

 

= lies on the incircle and these lines: {11, 30}, {12, 2222}, {55, 2687}, {56, 1290}, {65, 23341}, {513, 3028}, {517, 3024}, {523, 1317}, {942, 3025}, {1319, 1365}, {1325, 5172}, {1358, 1443}, {2646, 3326}, {3318, 10149}
= reflection of X(5520) in X(3109)
= reflection of X(3028) in the X(1)X(3) line

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Parallelogic A*B*C*, A1B1C1; 

= X(1)X(8674)∩X(11)X(11735)

= a*(2*a^9 - 2*a^8*b - 3*a^7*b^2 + 4*a^6*b^3 - 3*a^5*b^4 + 7*a^3*b^6 - 4*a^2*b^7 - 3*a*b^8 + 2*b^9 - 2*a^8*c + 4*a^7*b*c - 4*a^5*b^3*c + 4*a^4*b^4*c - 5*a^3*b^5*c - a^2*b^6*c + 5*a*b^7*c - b^8*c - 3*a^7*c^2 + 12*a^5*b^2*c^2 - 6*a^4*b^3*c^2 - 8*a^3*b^4*c^2 + 9*a^2*b^5*c^2 - a*b^6*c^2 - 3*b^7*c^2 + 4*a^6*c^3 - 4*a^5*b*c^3 - 6*a^4*b^2*c^3 + 14*a^3*b^3*c^3 - 4*a^2*b^4*c^3 - 5*a*b^5*c^3 + 3*b^6*c^3 - 3*a^5*c^4 + 4*a^4*b*c^4 - 8*a^3*b^2*c^4 - 4*a^2*b^3*c^4 + 8*a*b^4*c^4 - b^5*c^4 - 5*a^3*b*c^5 + 9*a^2*b^2*c^5 - 5*a*b^3*c^5 - b^4*c^5 + 7*a^3*c^6 - a^2*b*c^6 - a*b^2*c^6 + 3*b^3*c^6 - 4*a^2*c^7 + 5*a*b*c^7 - 3*b^2*c^7 - 3*a*c^8 - b*c^8 + 2*c^9) : : 

 

= 3 X[5603] - X[10767],3 X[15061] - X[19914] 

 

= lies on these lines: {1, 8674}, {11, 11735}, {21, 104}, {74, 10698}, {100, 7984}, {113, 11729}, {125, 952}, {214, 16598}, {1537, 2777}, {2773, 13868}, {2800, 11709}, {5603, 10767}, {5663, 19907}, {6224, 10778}, {7972, 13211}, {9140, 10031}, {10693, 17660}, {15061, 19914}

 

= midpoint of X(i) and X(j) for these {i,j}: {74, 10698}, {100, 7984}, {6224, 10778}, {7972, 13211}, {9140, 10031}, {10693, 17660}
= reflection of X(i) in X(j) for these {i,j}: {11, 11735}, {113, 11729}
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Best regards,
Peter Moses.