HYACINTHOS 27198

 [Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:

(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp. 

La, Lb, Lc = the radical axes of ((Nb),(Nc)), ((Nc),(Na)), ((Na),(Nb)), resp.

 

La1, Lb1, Lc1 = the reflections of La, Lb, Lc in BC, CA, AB, resp.

La2, Lb2, Lc2 = the parallels to La1, Lb1, Lc1 through A, B, C, resp.

La3, Lb3, Lc3 = the reflections of La2, Lb2, Lc2 in BC, CA, AB, resp.

La4, Lb4, Lc4 = the parallels to La3, Lb3, Lc3 through A, B, C, resp.

 

 La4, Lb4, Lc4 are concurrent.

or if A*B*C* = the triangle bounded by  La3, Lb3, Lc3,  then ABC, A*B*C* are parallelogic 

Parallelogic centers in terms of P?

 

[César Lozada]:

 

IF P= x: y :z (barycentrics), then

 

Q = A->A* = antigonal conjugate of P

 

Q* = A*->A =  (a^4-2*a^2*b^2-a^2*c^2+b^4+2* b^2*c^2)*c^4*x^3*y^2+(a^4-3*a^ 2*b^2-3*a^2*c^2+2*b^4+2*b^2*c^ 2+2*c^4)*b^2*c^2*x^3*y*z+(a^4- a^2*b^2-2*a^2*c^2+2*b^2*c^2+c^ 4)*b^4*x^3*z^2-(-a^2+b^2+c^2)* (a^2-b^2+c^2)*c^4*x^2*y^3+(a^ 6-2*a^4*b^2-a^2*b^4+a^2*b^2*c^ 2+2*b^6-3*b^4*c^2+2*b^2*c^4-c^ 6)*c^2*x^2*y^2*z+(a^6-2*a^4*c^ 2+a^2*b^2*c^2-a^2*c^4-b^6+2*b^ 4*c^2-3*b^2*c^4+2*c^6)*b^2*x^ 2*y*z^2-(a^2+b^2-c^2)*(-a^2+b^ 2+c^2)*b^4*x^2*z^3+(a^2-b^2+c^ 2)*(a^4-2*a^2*b^2+b^2*c^2-c^4) *c^2*x*y^3*z+(a^6*b^2+a^6*c^2- a^4*b^2*c^2-b^8+2*b^6*c^2-2*b^ 4*c^4+2*b^2*c^6-c^8)*x*y^2*z^ 2+(a^2+b^2-c^2)*(a^4-2*a^2*c^ 2-b^4+b^2*c^2)*b^2*x*y*z^3+(a^ 2-b^2+c^2)*(a^2*c^2+b^4-c^4)* a^2*y^3*z^2+(a^2+b^2-c^2)*(a^ 2*b^2-b^4+c^4)*a^2*y^2*z^3 : :

 

ETC-pairs (P, Q*):

(1,16110), (3,399), (30,1154), (67,9973), (265,6243), (511,5965), (512,826), (895,11008), (1176,11061), (1510,523), (1916,7877), (2777,10628), (3065,16014), (5504,9936), (6368,520), (9019,524), (11559,382), (13152,1510), (13754,539), (14366,14683)

 

For P on the circumcircle Q*(P) = X(4)

 

Some others:

Q*(X(2)) = X(316)X(2393) ∩ X(598)X(11188)

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

= on lines: {316, 2393}, {598, 11188}, {671, 2854}, {691, 2936}, {2386, 7799}, {2930, 14246}, {2979, 7850}, {4611, 9157}, {7812, 9971}, {8681, 11054}

= [ 1.1759262819242200, -13.0148587753110900, 12.1082161193116500 ]

 

Q*(X(6)) = midpoint of X(11008) and X(14683)

= (5*a^8-5*(b^2+c^2)*a^6-(3*b^4- 11*b^2*c^2+3*c^4)*a^4+5*(b^4- c^4)*(b^2-c^2)*a^2-2*(b^4-c^4) ^2 : : (barys)

= 3*X(6)-2*X(67), 5*X(6)-4*X(125), 3*X(6)-4*X(5095), 9*X(6)-8*X(15118), 5*X(67)-6*X(125), 3*X(67)-4*X(15118), 3*X(125)-5*X(5095), 9*X(125)-10*X(15118), 2*X(265)-3*X(5102), 3*X(599)-4*X(6593), 2*X(895)-3*X(15534), 5*X(2930)-6*X(9143), 3*X(5095)-2*X(15118), 3*X(5621)-4*X(8550), 3*X(9143)-5*X(11061)

= on lines: {6, 67}, {23, 524}, {265, 5102}, {382, 542}, {399, 5965}, {599, 6593}, {895, 8877}, {2781, 5925}, {2836, 3901}, {2854, 6144}, {3448, 3629}, {3520, 5621}, {5181, 15533}, {6698, 13169}, {9970, 15069}, {9973, 13417}, {10628, 10938}, {11008, 14683}

= midpoint of X(11008) and X(14683)

= reflection of X(i) in X(j) for these (i,j): (3448, 3629), (9973, 13417)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (67, 5095, 6), (67, 15140, 5094)

= [ -8.9965443314005800, -4.8721398416454050, 11.1659356021545400 ]

 

César Lozada