HYACINTHOS 23998

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Antreas P. Hatzipolakis

 

[APH]:

Let ABC be a triangle and A'B'C' the pedal triangle of I.

    Denote:

    H' = the orthocenter of A'B'C'

    Ab, Ac = the orth. projections of A on H'B', H'C', resp.

    L1 = te Euler line of AAbAc. Similarly L2, L3

[...]

Let Ia, Ib, Ic be the excenters and A1, B1, C1 their ortogonal projections on BC, CA, AB, resp.

The parallels to L1, L2, L3 through:

4. Ia, Ib, Ic, resp.

 

[..]

 

6. A*, B*, C* (where A*, B*, C* = the midpoints of AbAc, BcBa, CaCb, resp.)

APH

[Angel Montesdeoca]:

*** The parallels to L1, L2, L3 through   Ia, Ib, Ic, resp.  are concurrent in   X(2136) = X(145)-Ceva conjugate of X(1).

*** The parallels to L1, L2, L3 through   A*, B*, C* (where A*, B*, C* = the midpoints of AbAc, BcBa, CaCb, resp.)  are concurrent in  Z = 3(4R+r) X(7) + (4R-3r) X(8).
       Z = (a(3a^2(b+c)-2a b c-3b^3+5b c(b+c)-3c^3) : ... : ...),
 with (6-9-13)-search numbers (1.52416707476681, 1.01851923005744, 2.23207405735993)
       Z lies on lines: {1, 4004}, {2, 3922}, {7, 8}, {10, 3838}, {21, 5183}, {140, 517}, {226, 8256}, {354, 3623}, {528, 6738}, {758, 4662}, {942, 3244}, {946, 3847}, {958, 2093}, {960, 1698}, {999, 8668}, {1001, 7991}, {1155, 5303}, {1159, 3811}, {1357, 4767}, {1376, 3340}, {1836, 5554}, {2098, 3306}, {2802, 5045}, {2886, 4848}, {3057, 3622}, {3617, 3962}, {3626, 4757}, {3633, 5902}, {3649, 6735}, {3679, 4018}, {3698, 3740}, {3816, 4301}, {3826, 5837}, {3844, 5835}, {3872, 5221}, {3873, 3893}, {3876, 4731}, {3918, 5044}, {4002, 5692}, {4127, 4745}, {4711, 5904}, {5048, 5253}, {5439, 5697}, {5883, 9957}, {6928, 7686}.
      
       Z is the midpoint of X(i) and X(j) for these {i,j}: {65, 5836}, {960, 5903}, {3626, 4757}.
      
       Angel Montesdeoca