HYACINTHOS 27005

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

Oa, Ob, Oc = the reflections of O in BC, CA, AB, resp.

Oab, Oac = the reflections of Oa in AB, AC, resp.
Obc, Oba = the reflections of Ob in BC, BA, resp.
Oca, Ocb = the reflections of Oc in CA, CB, resp.

Ma, Mb, Mc = the midpoints of OabOac, ObcOba, OcaOcb, resp.

A*B*C* = the triangle bounded by OabOac, ObcOba, OcaOcb

A'B'C' = the orthic triangle of ABC
A"B"C" = the orthic triangle of MaMbMc

 

 

1. ABC, MaMbMc are orthologic.

 

2. A'B'C', A"B"C" are orthologic.

 

3. ABC, A*B*C* are orthologic.

 

4. A'B'C', A*B*C* are orthologic

 

5. A"B"C", A*B*C* are orthologic.

 

[César Lozada]:

 

 

      1) 1. (A <-> Ma) = (74 <-> 185)

2)     2. (A’ <-> A”) = (113 <-> 5446)

)      (3. (A <-> A*) = (54 <-> Q₃)

4)     4. (A’ <-> A*) = (6152 <-> 6242)

5)     5. (A” <-> A*) = (6152 <-> Q₅)

 

where:

 

Q₃ =midpoint of X(195) and X(382)

= (3*cos(2*A)+1)*cos(B-C)-cos(A) *cos(2*(B-C))-cos(A)-1/2*cos( 3*A) : : (trilinears)

= (R^2-SA)*S^2-(13*R^2-5*SW)*SB* SC : : (barycentrics)

= a^10-(5*b^4+b^2*c^2+5*c^4)*a^ 6+(b^2+c^2)*(5*b^4-7*b^2*c^2+ 5*c^4)*a^4-(b^4-c^4)*(b^2-c^2) ^3 : : (barycentrics)

 

= 3*X(3)-4*X(6689), 3*X(4)-X(2888), 4*X(4)-X(3519), 5*X(4)-X(12325), 3*X(51)-2*X(11802), 3*X(381)-2*X(1209), 3*X(381)-X(12307), 4*X(2888)-3*X(3519), 2*X(2888)-3*X(6288), 5*X(2888)-3*X(12325), 5*X(3519)-4*X(12325), 3*X(3574)-2*X(6689), 3*X(3830)+X(12316), X(6243)+2*X(12300), 5*X(6288)-2*X(12325)

 

= on lines: {3, 3574}, {4, 93}, {5, 7691}, {20, 10610}, {30, 54}, {51, 11802}, {52, 265}, {74, 11804}, {110, 11805}, {143, 3153}, {185, 10115}, {195, 382}, {381, 1209}, {389, 7574}, {399, 13419}, {539, 3830}, {550, 8254}, {567, 12225}, {1199, 13470}, {1478, 13079}, {1493, 3146}, {1531, 6153}, {1594, 3581}, {1657, 11425}, {2917, 7517}, {3091, 13565}, {3518, 14643}, {3583, 7356}, {3585, 6286}, {3627, 7728}, {5073, 10619}, {5076, 5965}, {5655, 7540}, {5893, 9935}, {6221, 8995}, {6240, 11597}, {6284, 10066}, {6398, 13986}, {7354, 10082}, {10274, 13352}, {10540, 11819}, {11750, 15087}, {12173, 15091}, {12208, 14880}, {12234, 13403}, {12606, 15739}, {13371, 15061}, {14855, 14861}

 

= midpoint of X(i) and X(j) for these {i,j}: {195, 382}, {3146, 12254}

 

= reflection of X(i) in X(j) for these (i,j): (3, 3574), (20, 10610), (74, 11804), (110, 11805), (185, 10115), (550, 8254), (3519, 6288), (6288, 4), (7691, 5), (12121, 11597), (12254, 1493), (12307, 1209), (12606, 15739)

 

= X(i)-of triangle-T for these (i, T): (3574, X3-ABC reflections), (7691, Johnson)

 

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (381, 12307, 1209), (3583, 7356, 12956), (3585, 6286, 12946)

 

= [ -18.6467413575916000, -1.6717442959076700, 13.4042142364242700 ]

 

Q₅ = reflection of X(54) in X(195)

= (4*cos(2*A)+1)*cos(B-C)+cos(3* A) : : (trilinears)

= (SB+SC)*(-3*(S-SA)*(S+SA)-(8* R^2-2*SW)*SA) : : (barycentrics)

 

= 3*X(2)-4*X(12242), 2*X(3)-3*X(54), X(3)-3*X(195), 4*X(3)-3*X(7691), 5*X(3)-6*X(10610), 5*X(3)-3*X(12307), X(3)+3*X(12316), 3*X(54)-4*X(1493), 5*X(54)-4*X(10610), 5*X(54)-2*X(12307), X(54)+2*X(12316), 3*X(195)-2*X(1493), 4*X(195)-X(7691), 5*X(195)-2*X(10610), 5*X(195)-X(12307), 8*X(1493)-3*X(7691), 5*X(1493)-3*X(10610), 10*X(1493)-3*X(12307), 2*X(1493)+3*X(12316), 3*X(2979)-4*X(12363), 5*X(7691)-8*X(10610), 5*X(7691)-4*X(12307), X(7691)+4*X(12316), 2*X(10610)+5*X(12316), X(12307)+5*X(12316)

 

= on the cubic K633 and these lines: {2, 11431}, {3, 54}, {4, 539}, {5, 1173}, {6, 11444}, {20, 10619}, {49, 12107}, {52, 110}, {61, 10677}, {62, 10678}, {74, 15089}, {113, 13420}, {155, 3060}, {156, 12380}, {193, 576}, {323, 389}, {381, 13432}, {394, 15043}, {511, 12226}, {524, 13160}, {546, 6288}, {568, 15091}, {569, 13472}, {578, 11004}, {632, 8254}, {895, 6145}, {973, 1995}, {1092, 15053}, {1181, 11577}, {1199, 1216}, {1209, 3090}, {1351, 5198}, {1614, 6243}, {1656, 12834}, {1992, 6816}, {1994, 5562}, {2070, 9705}, {2293, 3746}, {3146, 5878}, {3153, 10112}, {3303, 13079}, {3462, 14918}, {3525, 6689}, {3529, 12254}, {3555, 5887}, {3580, 3628}, {3627, 7728}, {4994, 14978}, {5056, 15004}, {5079, 13565}, {5563, 7356}, {5609, 14668}, {5640, 9827}, {5888, 15037}, {5899, 13421}, {6403, 9925}, {6419, 12965}, {6420, 12971}, {6759, 9935}, {7488, 9706}, {7507, 8537}, {7530, 13423}, {7545, 13368}, {7566, 15069}, {7574, 11264}, {7730, 13861}, {7991, 9905}, {7999, 13154}, {8718, 13391}, {9140, 13371}, {9716, 10274}, {9781, 15068}, {9924, 11477}, {10263, 14157}, {10540, 14449}, {10625, 15032}, {10628, 12086}, {11403, 12164}, {11432, 15028}, {11440, 13352}, {11472, 12111}, {11591, 14627}, {11597, 15034}, {11804, 15027}, {13383, 15360}, {13419, 14683}, {13482, 14130}, {13754, 14865}, {15022, 15605}, {15246, 15606}

 

= midpoint of X(i) and X(j) for these { i, j}: {4, 11271}, {195, 12316}

 

= reflection of X(i) in X(j) for these (i, j): (3, 1493), (5, 11803), (20, 10619), (54, 195), (74, 15089), (110, 2914), (2888, 3574), (3519, 5), (6242, 52), (7691, 54), (9935, 6759), (9972, 576), (11271, 13431), (11412, 12606), (12111, 12300), (12280, 6152), (12307, 10610), (12325, 1209)

 

= X(i)-of triangle-T for these (i, T): (1493, X3-ABC reflections), (3519, Johnson), (11271, Euler)

 

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i, j, k): (3, 195, 1493), (3, 1493, 54), (3, 11423, 5012), (3, 12161, 11423), (1614, 6243, 15107), (1993, 12160, 5889), (1994, 5562, 13434), (3060, 12280, 6152), (11126, 11127, 97), (11412, 11423, 3), (11412, 12161, 5012), (11432, 15066, 15028), (12161, 12606, 54)

 

= [ -32.1280029457988400, 20.4732108444645000, 4.2952121799545780 ]

 

César Lozada