HYACINTHOS 25271

[Antreas P. Hatzipolakis]:

 

Let ABC  be a triangle.

Denote:

A', B', C' = the reflections of N in BC, CA, AB, resp.
A", B", C" = the orthogonal projections of A, B, C on B'C', C'A', A'B', resp.

Ab, Ac = the orthogonal projections of A" on NB', NC', resp.
Bc, Ba = the orthogonal projections of B" on NC', NA', resp.

Ca, Cb = the orthogonal projections of C" on NA', NB', resp.

La, Lb, Lc = the Euler lines of A"AbAc, B"BcBa, C"CaCb, resp.

A*B*C* = the triangle bounded by La, Lb, Lc

ABC, A*B*C* are parallelogic.

The parallelogic center (ABC, A*B*C*) is de Longchamps point X20 (reflection of H in O)

The other one?

[César Lozada]:

 

Z(A*->A) = complement of X(389)

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

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

= 3*X(2)-X(389)  = (12*R^2-SW)*X(3)-(4*R^2-SW)*X( 64) = (4*R^2-SW)*X(5)+SW*X(141)

= On lines: {2,389}, {3,64}, {4,3917}, {5,141}, {20,7998}, {24,5651}, {30,5447}, {51,3090}, {52,1656}, {54,3292}, {68,5486}, {69,6804}, {140,9729}, {143,547}, {155,182}, {184,7509}, {185,631}, {264,8887}, {297,6750}, {373,3567}, {376,11381}, {381,10625}, {394,578}, {546,10627}, {549,5876}, {568,5070}, {632,5892}, {916,9940}, {960,2818}, {970,6911}, {1092,7503}, {1147,7514}, {1154,3628}, {1181,7484}, {1209,2072}, {1350,1598}, {1352,6643}, {1364,3074}, {1495,7512}, {1872,5784}, {2807,6684}, {2979,3091}, {3060,5056}, {3075,7066}, {3089,10519}, {3098,7387}, {3313,10516}, {3524,6241}, {3525,5890}, {3526,9730}, {3527,11477}, {3530,5663}, {3537,6225}, {3781,5709}, {3784,7330}, {3861,11017}, {4260,5707}, {5055,6243}, {5071,9781}, {5092,7516}, {5640,7486}, {5752,6918}, {5777,11573}, {5878,10996}, {5946,10219}, {6826,10441}, {7485,10984}, {7487,20002}, {10303,10574}

= midpoint of X(i) and X(j) for these {i,j}: {3,5907}, {5,1216}, {140,11591}, {389,5562}, {546,10627}, {1352,11574}, {3819,5891}, {5446,6101}, {5777,11573}

= reflection of X(i) in X(j) for these (i,j): (389,11695), (3861,11017), (5462,3628), (5946,10219), (9729,140), (10110,5)

= anticomplement of X(11695)

= complement of X(389)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2,389,11695), (2,5562,389), (2,11444,5562), (3,5891,5907), (3,9306,10282), (4,7999,3917), (5,6101,5446), (52,1656,5943), (155,7393,182), (185,5650,631), (394,7395,578), (631,11459,185), (632,6102,5892), (1092,7503,11430), (1216,5446,6101), (1216,10170,5), (3090,11412,51), (3567,5067,373), (3628,5462,6688), (3819,5907,3), (6643,11487,1352), (7485,11441,10984)

= [ 2.434429148970821, 1.11369786717613, 1.746060197414374 ]

 

César Lozada