HYACINTHOS 23972

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Antreas P. Hatzipolakis

 

[APH]:

Let ABC be a triangle and A'B'C', A"B"C" the pedal and cevian triangles of I, resp..

    Denote:

    Ab, A2 = the orthogonal projections of A" on IB, IB', resp.
    Ac, A3 = the orthogonal projections of A" on IC, IC', resp.

    (Nab), (Nac) = the NPCs of A"AbA2, A"AcA3, resp.

    R1 = the radical axis of (Nab), (Nac) [it is the perp. bisector of NabNac]

    Similarly R2, R3.

    R1, R2, R3 are concurrent.

    The parallels to R1, R2, R3 through A', B', C', resp. are concurrent.
    (on the OI line)

    The parallels to R1, R2, R3 through A",B",C" resp. are concurrent.

[Angel Montesdeoca]:

*** R1, R2, R3 are concurrent in
W=( a (a^7 (b-c)^2-6 a^6 b c (b+c)+b (b-c)^4 c (b+c)^3-2 a^2 b (b-c)^2 c (4 b^3+9 b^2 c+9 b c^2+4 c^3)+a^4 b c (13 b^3+9 b^2 c+9 b c^2+13 c^3)+a^5 (-3 b^4+3 b^3 c-16 b^2 c^2+3 b c^3-3 c^4)-a (b^2-c^2)^2 (b^4+b^3 c+2 b^2 c^2+b c^3+c^4)+3 a^3 (b^6+5 b^4 c^2+4 b^3 c^3+5 b^2 c^4+c^6)),b (a^8 (-b+c)-a^7 c (b+c)+b c^2 (b^2-c^2)^3+a^5 c^2 (-2 b^2+b c+3 c^2)+a^6 (3 b^3-8 b^2 c-3 c^3)+a c (b^2-c^2)^2 (-2 b^3-6 b^2 c-b c^2+c^3)+a^3 c (3 b^5+9 b^4 c+12 b^3 c^2+10 b^2 c^3+b c^4-3 c^5)+a^4 (-3 b^5+13 b^4 c+15 b^3 c^2+10 b^2 c^3+2 b c^4+3 c^5)+a^2 (b^7-6 b^6 c-16 b^5 c^2+9 b^4 c^3+15 b^3 c^4-2 b^2 c^5-c^7)): ... : ....)

with (6-9-13)-search number  (2.51895233670473, 2.46696698017884, 0.770170878689158)

On lines: {73,500}, {511,9940}

*** The parallels to R1, R2, R3 through A', B', C', resp. are concurrent in X(942) = inverse in incircle of X(36)

***    The parallels to R1, R2, R3 through A",B",C" resp. are concurrent in X(500) = orthocenter of the incentral triangle

Angel Montesdeoca