HYACINTHOS 25933

[Antreas P. Hatzipolakis]:

Let ABC be a triangle, A'B'C' the pedal triangle of H and P a point on the Euler line.

    Denote:

    Pa, Pb, Pc = the orthogonal projections of P on HA', HB', HC', resp.

    Pab, Pac = the orthogonal projections of Pa on AC, AB, resp.

    La = the Euler line of APabPac, Similarly Lb, Lc

    L1 = the Euler line of PaPabPac. Similarly L2, L3.

 

[Angel Montesdeoca]:

*** 1. The locus of  point of concurrence W1 of  lines La, Lb, Lc, as P moves on the Euler line, is the line  d1 that passes through X(i), por i=125, 403, 6000, 12292.
 Pairs {P, W1}:  {4,125}, {3146,12292}.
 
***    2. The parallels to La, Lb, Lc through A, B, C are concurrent at X(74), for all P on the Euler line.
  
***    3. The parallels to La, Lb, Lc through A', B', C' are concurrent at X(1986) on the line d4, for all P on the Euler line.

***     4. The locus of  of point of concurrence W4 of  lines L1, L2, L3, as P moves on the Euler line, is the line  d4 that passes through X(i), por i=125, 389, 1594, 1986, 3574, 10628, 12300.
 Pairs {P, W4}:  {3,389}, {4,3574}, {3146,12300}.
 
***     5. The parallels to L1, L2, L3 through A, B, C are concurrent at X(54)=Kosnta Point, for all P on the Euler line.
    
***     6. The parallels to L1, L2, L3 through A', B', C' are concurrent  at X(6152)=orthologic center of the pedal triangles of X(4) and X(5), for all P on the Euler line.
    
    
*** The envelope of the line W1W4, when P traverses the line of Euler, is the parabola tangent to d1 at the  reflection of X(10540) = X(3)X(64)/\X(4)X(49) in X(6699) =  centroid of {A,B,C,X(74)}, to d4 in X(3574) = midpoint of X(4) and X(54) , and its axis has the direction of X(3)X(54).

With these information (It_pt_p), this parabola can be constructed (§12.3. Parabola1tangent1tangent-at, axis-direction).

The point of tangency  of d1 with the parabola (reflection of X(10540) in X(6699) ) is:

T1 = (2 a^8 (b^2+c^2)-a^6 (7 b^4-10 b^2 c^2+7 c^4)+9 a^4 (b^2-c^2)^2 (b^2+c^2)-5 a^2 (b^4-c^4)^2+(b^2-c^2)^4 (b^2+c^2) : ... : ...),

T1 lies on lines: {30,6070}, {74,10421}, {125,403}, {185,427}, {542,2071}, {1533,10264}, {1568,5663}, {1596,11381}, {3357,11457}, {3520,10619}, {5642,10257}, {6241,7577}, {6353,12324}, {6699,10540}, {10112,12086}, {10193,11464}, {10605,11550}.

with (6,9,13)-search numbers {12.6437147591342, 13.1454225017087, -11.2955732927221}.

Angel Montesdeoca