HYACINTHOS 29002

[Antreas P. Hatzipolakis]:

Let ABC be a triangle, P a point and A'B'C' the antipedal triangle of P.

Denote:

(Obc), (Ocb) = the circles (B, BC'), (C, CB'), resp.

(Oca), (Oac) = the circles (C, CA'), (A, AC'), resp.  

(Oab), (Oba) = the circles (A, AB'), (B, BA'), resp.    

Ra = the radical axis of  (Obc), (Ocb) 

Rb = the radical axis of  (Oca), (Oac) 
Rc = the radical axis of  (Oab), (Oba) 

H, O, I lie on the locus of P such that Ra, Rb, Rc are concurrent.

For 
1. P = H, point of concurrence is X(20)
2. P = O, point ? 

3. P = I, point ?

[Peter Moses]:


Hi Antreas,

2). = MIDPOINT OF X(9914) AND X(12085)

 

=  a^2*(a^14 - 3*a^12*b^2 + a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8 - a^4*b^10 + 3*a^2*b^12 - b^14 - 3*a^12*c^2 + 10*a^10*b^2*c^2 - 13*a^8*b^4*c^2 + 8*a^6*b^6*c^2 - a^4*b^8*c^2 - 2*a^2*b^10*c^2 + b^12*c^2 + a^10*c^4 - 13*a^8*b^2*c^4 + 2*a^6*b^4*c^4 + 2*a^4*b^6*c^4 + 5*a^2*b^8*c^4 + 3*b^10*c^4 + 5*a^8*c^6 + 8*a^6*b^2*c^6 + 2*a^4*b^4*c^6 - 12*a^2*b^6*c^6 - 3*b^8*c^6 - 5*a^6*c^8 - a^4*b^2*c^8 + 5*a^2*b^4*c^8 - 3*b^6*c^8 - a^4*c^10 - 2*a^2*b^2*c^10 + 3*b^4*c^10 + 3*a^2*c^12 + b^2*c^12 - c^14) : : 

 

= (1 + J^2) X[3] - 2 X[64], (7 - J^2) X[3] - 6 X[154], (7 - J^2) X[64] - 3 (1 + J^2) X[154].

 

= lies on these lines: {3, 64}, {4, 18532}, {22, 9833}, {24, 14216}, {25, 13419}, {26, 1503}, {30, 9938}, {159, 2918}, {161, 2937}, {186, 12324}, {265, 7517}, {378, 5878}, {511, 9908}, {542, 9937}, {1216, 1660}, {1593, 18388}, {1598, 18383}, {1853, 7506}, {2071, 12250}, {2777, 9914}, {2781, 16266}, {2883, 7526}, {3515, 13171}, {3517, 14864}, {3518, 32064}, {3520, 6225}, {3818, 6642}, {5073, 9919}, {5198, 18376}, {5656, 14118}, {5893, 31861}, {6053, 13293}, {6247, 6644}, {6293, 18445}, {6640, 32125}, {7387, 12293}, {7512, 11206}, {7514, 16252}, {7516, 10192}, {7525, 15577}, {7529, 23325}, {7555, 15581}, {9934, 12292}, {10249, 15805}, {10628, 12164}, {11413, 20427}, {12083, 17845}, {12084, 15311}, {22658, 22978}

 

= midpoint of X(9914) and X(12085)

= crosspoint of X(1288) and X(15384)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1619, 6759}, {64, 1498, 18439}, {3357, 6759, 5907}

3). X(84).

Best regards,
Peter Moses.