HYACINTHOS 25755

[Tran Quang Hung]:
    
    Let ABC be a triangle.
    A',B',C' are midpoints of BC,CA,AB, reps.
    A'' is projection of A on B'C'.
    Circle (A,AA'') cuts the segments CA,AB at Ac,Ab.    da is the line passing through Ac,Ab.

    Define similarly, we have line db,dc.

    The lines da,db,dc bound a triange which is perspective to ABC.
    Which is the perspector ?

 

[Angel Montesdeoca]:

*** The lines da,db,dc bound a triange which is perspective to ABC, with perspector the incenter.

 Define similarly Ba, Bc, Cb, Ca.
 
 The center of the conic that passes through Ac ,Ab,  Ba, Bc, Cb, Ca is
 
 Wi = ( a(a(b+c)-b^2-c^2 + 4R(b+c-a)) : ... : ...),
 
 with (6-9-13)-search numbers (2.68394388667284, 1.97661022831583, 1.03349868383980).
 On the lines: {1, 6}, {4662, 7090}, {6212, 9943}.
 
****  Let A'c, A'b be the reflections of Ac, Ab in A.  Define similarly B'a, B'c, C'b, C'a.
 The lines A'cA'b, B'aB'c,  C'bC'a bound a triange which is perspective to ABC, with perspector the incenter.
 
 The center of the conic that passes through A'c ,A'b,  B'a, B'c, C'b, C'a is
 
 We = ( a(a(b+c)-b^2-c^2 -  4R(b+c-a)) : ... : ...),
 
 with (6-9-13)-search numbers  (4.46000151178707, 2.48835564572162, -0.140505509032314).
 On the lines:  {1, 6}, {3812, 7090}, {6213, 9943}.
 
 The midpoint of Wi and We is X(9).
 
 Angel Montesdeoca