HYACINTHOS 29227

[Antreas P. Hatzipolakis]:

 

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

Denote:

D =  the Feuerbach point X(11)

A", B", C" = the reflections of A', B', C' in OI line.

D' = the orthogonal projection of D on the OI line.
D" = the reflection of D in the OI line

1. The NPCs of DD'A", DD'B", DD'C" are coaxial.

2. The NPCs of D'D"A', D'D"B', D'D"C' are coaxial.

2nd (other than the midpoints of DD', D'D") intersections?

 

[César Lozada]:

 

1)      The 2nd point of intersection, other than the midpoint (D,D’), is Q12=X(12736).

The midpoint of (D, D’) is:

Q11 = COMPLEMENT OF X(15632)

= a*(-a+b+c)*(b-c)^2*(a^5-(2*b-c)*(b-2*c)*a^3-3*(b+c)*b*c*a^2+(b^4+c^4-b*c*(5*b^2-12*b*c+5*c^2))*a+3*(b^2-c^2)*(b-c)*b*c) :: (barys)

= 3*X(11)+X(3025), X(3025)-3*X(14115), 9*X(16173)-X(23153)

= lies on these lines: {2, 15632}, {11, 513}, {244, 6129}, {517, 1387}, {650, 2170}, {1086, 3326}, {1149, 5048}, {3011, 18839}, {3022, 14027}, {3086, 31849}, {3660, 11028}, {15608, 21252}, {16173, 23153}

= midpoint of X(i) and X(j) for these {i,j}: {11, 14115}, {3259, 15635}

= complement of X(15632)

= X(3233)-of-intouch triangle

= [ 2.8170022456122520, 3.0684830776965950, 0.2161751609118500 ]

 

2)      The 2nd point of intersection, other than the midpoint (D’, D”), is:

Q22 = MIDPOINT OF X(65) AND X(3025)

= a*( (b+c)*a^8-(3*b^2+2*b*c+3*c^2)*a^7-2*(b+c)*(b^2-5*b*c+c^2)*a^6+(9*b^4+9*c^4-2*b*c*(4*b^2+5*b*c+4*c^2))*a^5-(b+c)*(16*b^2-33*b*c+16*c^2)*b*c*a^4-(9*b^4+9*c^4-b*c*(b+4*c)*(4*b+c))*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(2*b^4+2*c^4+b*c*(6*b^2-17*b*c+6*c^2))*a^2+(b^2-c^2)^2*(3*b^2-3*b*c+c^2)*(b^2-3*b*c+3*c^2)*a-(b^2-c^2)^3*(b-c)^3) : : (barys)

= 3*X(354)-X(13756), 5*X(18398)-X(23153)

= lies on these lines: {1, 901}, {57, 953}, {65, 3025}, {226, 31841}, {354, 13756}, {513, 12736}, {517, 5083}, {942, 24201}, {1210, 3259}, {2800, 14115}, {13411, 22102}, {18398, 23153}

= midpoint of X(65) and X(3025)

= reflection of X(24201) in X(942)

= inverse in incircle of X(2718)

= (inverse-in-incircle)-isogonal conjugate of-X(952)

= (intouch)-complement of-X(13756)

= X(477)-of-inverse-in-incircle triangle

= X(25641)-of-intouch triangle

= X(31379)-of-Ursa-minor triangle

= [ 1.9341649661092270, 1.4438330630791940, 1.7483962231099090 ]

 

The midpoint of (D’,D”) is:

Q21 = MIDPOINT OF X(3025) AND X(14115)

= a*(b-c)^2*(-a+b+c)*(3*a^5-(6*b^2-7*b*c+6*c^2)*a^3-(b+c)*b*c*a^2+(3*b^4+3*c^4-b*c*(7*b^2-12*b*c+7*c^2))*a+(b^2-c^2)*(b-c)*b*c) : : (barys)

= X(11)+3*X(3025), X(11)-3*X(14115)

= lies on these lines: {11, 513}, {517, 5083}, {1155, 1458}, {1357, 3328}, {4162, 7004}, {4293, 31849}

= midpoint of X(3025) and X(14115)

= [ 3.1647364943463510, 2.4599481424119230, 0.4769758474624240 ]

 

César Lozada