HYACINTHOS 26992

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C', A"B"C" the cevian triangles of H, G, resp.

Denote:

A1B1C1 = the midheight triangle
(ie A1, B1, C1 = the midpoints of AA', BB', CC', resp.)

A2, B2, C2 = the reflections of A1, B1, C1 in H, resp.

A3, B3, C3 = the reflections of A2, B2, C2 in BC, CA, AB, resp.

 

1. A3B3C3, A"B"C" are pespective.

 

2. The parallels to A3A", B3B", C3C", through A, B, C, resp. are concurrent.

 

3. The parallels to A3A", B3B", C3C", through A', B', C', resp. are concurrent.

 

4. The parallels to A3A", B3B", C3C", through A1, B1, C1, resp. are concurrent.

 

5. The parallels to A3A", B3B", C3C", through A2, B2, C2, resp. are concurrent.

 

[César Lozada]:

 

Hi Antreas,

 

1)      Q1 = X(6409)X(10962) ∩ X(6410)X(10960)

= (SB+SC)*(S^2-4*SB*SC)*(4*SA^2- S^2) : : (barys)

= on lines: {6, 8912}, {110, 3532}, {113, 1657}, {141, 3523}, {376, 2883}, {960, 7987}, {1147, 1192}, {1209, 5054}, {1498, 11598}, {1620, 12164}, {5023, 11672}, {6409, 10962}, {6410, 10960}, {8542, 10541}

= [ 4.1423623039329250, 2.8833749115280530, -0.2673776740425469 ]

 

2)      Q2 = X(54)X(3545) ∩ X(64)X(3543)

= SA*(4*SB^2-S^2)*(4*SC^2-S^2) : : (barys)

= on Jerabek hyperbola and these lines: {6, 1131}, {54, 3545}, {64, 3543}, {3426, 3853}, {3431, 5067}, {3527, 3845}, {3532, 5059}, {5056, 14528}, {11001, 11270}, {11744, 12324}, {13851, 15077}

= [ -0.3966182303996964, -0.5080122380786714, 4.1754198299925500 ]

 

Q2 is a point on the Jerabek hyperbola. Its isogonal conjugate Q2^-1 lies on the Euler line:

 

Q2^-1 = circumcircle-inverse-of X(13473)

= SB*SC*(4*SA^2-S^2) :: (barys)

= 5*(4*R^2-SW)*X(3)+2*R^2*X(4), X(4)-5*X(3147)

= As a point on the Euler line, this center has Shinagawa coefficients (-5*F, E+5*F)

= on lines: {2, 3}, {64, 1495}, {74, 12315}, {154, 1204}, {165, 11363}, {184, 1192}, {187, 3172}, {232, 5023}, {1112, 15051}, {1151, 5411}, {1152, 5410}, {1181, 11202}, {1350, 11470}, {1398, 5204}, {1829, 7987}, {1968, 5210}, {1986, 15040}, {2207, 5206}, {2931, 12301}, {3167, 11449}, {3199, 8588}, {3576, 11396}, {5010, 11399}, {5024, 10312}, {5050, 8537}, {5085, 12167}, {5090, 10164}, {5217, 7071}, {5412, 6410}, {5413, 6409}, {5894, 15448}, {6221, 10881}, {6241, 14530}, {6398, 10880}, {6403, 12017}, {6411, 11473}, {6412, 11474}, {6459, 13937}, {6460, 13884}, {6746, 15045}, {7280, 11398}, {8273, 11383}, {8541, 10541}, {8567, 11381}, {8722, 11380}, {8780, 12111}, {9777, 11425}, {9786, 11402}, {10282, 10605}, {10902, 11401}, {11270, 12112}, {11408, 11481}, {11409, 11480}, {13093, 14157}, {13148, 15034}, {13366, 14528}, {15105, 15152}

= circumcircle-inverse-of X(13473)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3, 4, 11410), (3, 25, 3516), (3, 1598, 3520), (3, 2070, 12085), (3, 3517, 378), (3, 9714, 12084), (3, 9909, 11413), (24, 1593, 25), (25, 3516, 11403), (26, 15646, 3), (1113, 1114, 13473), (1593, 3515, 24), (1598, 3520, 1593), (3517, 5198, 25), (3522, 6353, 1885), (7488, 15078, 3)

 

= [ 4.0137326199531500, 3.1336243690586640, -0.3812597520346106 ]

 

 

3)      Q3 = X(2)X(5893) ∩ X(4)X(8780)

= (S^2-4*SB*SC)*(2*SA-16*R^2+3* SW) : : (barys)

= on lines: {2, 5893}, {4, 8780}, {6, 1131}, {20, 5972}, {52, 6623}, {185, 3091}, {11439, 15431}, {12111, 15010}

= [ -2.8050269610881190, -3.9699790025613410, 7.6837393118752090 ]

 

4)      Q4 = X(6)X(1131) ∩ X(381)X(12242)

= (32*R^2-3*SA-5*SW)*S^2+32*(4* R^2-SW)*SB*SC : : (barys)

= on lines: {6, 1131}, {381, 12242}, {382, 7687}, {389, 3843}, {974, 12290}

= [ -1.6008225957439080, -2.2389956203200050, 5.9295795709338790 ]

 

5)      Z5 = (3*SA-80*R^2+17*SW)*(S^2-4*SB* SC) : : (barys)

= on lines {}

= [ -9.7524162261091670, -10.8233329166507300, 15.6348562977929600 ]

 

I wish you a happy new year and plenty of health and prosperity.

 

Best regards,

César Lozada