HYACINTHOS 28830

[Nguyễn Đăng Khoa]:

 

Let ABC be a triangle with circumcircle (O) and orthocenter H. Denote A’ the reflection of H in A. The parallel to BC through A cuts (O) in A” and the pole of this line w/r to (O) is A*.

Build B’, C’, B”, C”, B*, C* cyclically.

Then

1)      A’A”, B’B”, C’C” are concurrent

2)      A’A*, B’B*, C’C* are concurrent.

 

See AOPS (nguyendangkhoa17112003) .

 

[César Lozada]:

 

1)      A”B”C” is named the 1^st anti-circumperp triangle in the ETC index of triangles.  Lines A’A”, B’B”, C’C” concur at X(11411)

 

2)      A* has barycentric coordinates: - a^2*(b^2+c^2)  :  b^2*(b^2-c^2) : c^2*(c^2-b^2).

 

A’A*, B’B*, C’C* concur at:

Q1 = X(3)X(49) ∩ X(24)X(157)

= (S^2-SB*SC)*(S^2+2*R^2*(4*R^2+SA-4*SW)+SW^2) : : (barys)

= lies on these lines: {3, 49}, {4, 19172}, {24, 157}, {25, 8887}, {26, 2351}, {68, 23181}, {418, 6146}, {578, 16035}, {973, 23635}, {1624, 7505}, {2917, 7669}, {2937, 13558}, {3133, 12134}, {3135, 9833}, {6776, 26876}, {7512, 8266}, {7592, 15231}, {8553, 17849}, {9715, 15512}, {18925, 26874}

= [ -6.3665149060388020, -11.9618086564776300, 14.8603081237944900 ]

 

Triangle A*B*C* is perspective with perspector X(3) to the following triangles: anti-Hutson intouch, anti-incircle-circles, 6th anti-mixtilinear, Ara, Ascella, 1st Brocard-reflected, 1st Brocard, 1st circumperp, 2nd circumperp, Ehrmann-side, 1st Ehrmann, 2nd Euler, inner-Fermat, outer-Fermat, Fuhrmann, 2nd Fuhrmann, Johnson, K798e, K798i, Kosnita, McCay, medial, inner-Napoleon, outer-Napoleon, 1st Neuberg, 2nd Neuberg, tangential, Trinh, inner-Vecten, outer-Vecten. Also, A*B*C* is perspective to the following  triangles with given perspectors: (anti-inverse-in-incircle, 160), (anticomplementary, 8266), (5th Euler, 25), (anti-Euler, Q1), (Steiner, 1634) and 1^st excosine with perspector:

 

Q2 = X(20)X(64) ∩ X(154)X(160)

= (SB+SC)*(3*S^4-(4*R^2*(8*R^2+4*SA-3*SW)-4*SA^2+2*SB*SC+SW^2)*S^2+2*(4*R^2-SW)*SB*SC*SW) : : (barys)

= 3*X(154)-4*X(160)

= lies on these lines: {6, 1987}, {20, 64}, {154, 160}, {6748, 17810}, {18445, 22552}

= [ -16.2164662445990600, -24.3191834501926600, 27.9615451372404000 ]

 

César Lozada