HYACINTHOS 28398

 

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, P a point and A1B1C1, A2B2C2 the pedal, antipedal triangles of P, resp.

Denote:

D = the Poncelet point of ABCP

For P = I (D = X(11)):

1. The parallels to DA1, DB1, DC1 through A2, B2, C2, resp. are concurrent.

2. The parallels to DA2, DB2, DC2 through A1, B1, C1, resp. are concurrent.

Points?

Loci (if not complicated :)  ?

 

[César Lozada]:

 

 

1)

Loci = {Linf} ∪ {circumcircle} ∪ {q4: circum-quartic through ETC’s X(4)} ∪ {q8: degree-8 circum-excentral-curve through vertices of CIRCUMTANGENTIAL triangle and ETC’s 1, 4}

 

q4: ∑ [y*z*((2*SA^2-2*S^2)*x^2+a^4*y*z)] = 0

 

q8: ∑ [y*z*((-(a^4+b^4+2*b^2*c^2+3*c^4-2*(b^2+2*c^2)*a^2)*b^2*c^2*y-(a^4+3*b^4+2*b^2*c^2+c^4-2*(2*b^2+c^2)*a^2)*b^2*c^2*z)*x^5+(-c^2*(a^2+2*b^2-2*c^2)*(a^4-2*(b^2+c^2)*a^2+b^4+c^4)*y^2+2*b^2*c^2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*z*y-b^2*(a^2-2*b^2+2*c^2)*(a^4-2*(b^2+c^2)*a^2+b^4+c^4)*z^2)*x^4+a^2*(2*a^6-3*(b^2+c^2)*a^4+8*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2))*z^2*y^2*x^2-2*a^6*y^4*z^2*c^2+2*a^6*(a^2-b^2-c^2)*z^3*y^3-2*a^6*y^2*z^4*b^2)] = 0

 

For P=I the point of concurrence is Q1(X(1)) = X(1768)

 

2)

Loci = {Linf} ∪ {circumcircle} ∪ {q4 given above} ∪ {q12: degree-12 circum-excentral-curve through ETC’s 1, 4}

q12: Complicated equation

 

For P=I the point of concurrence is

 

Q2(X(1)) =  X(7)X(100) ∩ X(11)X(57)

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

= X(11)+3*X(11246), 3*X(65)+X(18976), 3*X(553)-X(5083)

= on lines: {5, 7702}, {7, 100}, {11, 57}, {30, 18838}, {56, 1387}, {65, 952}, {80, 3339}, {109, 1086}, {149, 21454}, {214, 3671}, {226, 3035}, {388, 1145}, {516, 3660}, {528, 553}, {938, 10724}, {942, 5840}, {1155, 5762}, {1317, 3340}, {1320, 3600}, {1466, 10090}, {1470, 11729}, {1537, 4295}, {1617, 3474}, {2099, 11046}, {2802, 4298}, {2829, 4292}, {2834, 3937}, {3036, 4848}, {3218, 5857}, {3333, 14217}, {3336, 8068}, {3337, 5533}, {3361, 16173}, {3911, 5087}, {4355, 5541}, {4440, 14594}, {4654, 6174}, {4860, 13274}, {5221, 12019}, {5708, 10738}, {5854, 10106}, {6147, 10044}, {7972, 18421}, {9945, 12739}, {10404, 10956}, {10742, 18541}, {11529, 12119}, {14151, 20095}, {15803, 21154}, {17579, 18419}, {17724, 23703}

= midpoint of X(4292) and X(12736)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5221, 13273, 12832), (12832, 13273, 12019)

= [ 0.6383105427686622, 0.9237167375291875, 2.7065634131094040 ]

 

César Lozada