HYACINTHOS 28783

[Antreas P. Hatzipolakis]:

 

Locus:

Let ABC be a triangle and P a point.

Denote:
(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.
(N1), (N2), (N3) = the reflections of (Na), (Nb), (Nc) in AP, BP, CP, resp.

R1, R2, R3 = the redical axes of ((N2), (N3)), ((N3), (N1)), ((N1), (N2)), resp.

Which is the locus of P such that the reflections of R1, R2, R3 in BC, CA, AB, resp. are concurrent ?

O lies on the locus.

Point ? (on the line at infinity)

G lies on the locus.
Point?

[César Lozada]:

 

Locus = {Vertices of ABC} ∪ {circum-degree-10} ∪ {q5=circum-degree-5 through vertices of medial triangle and ETC’s 2, 3, 4, 671, 1113, 1114 }

q5: ∑ [y*z*(2*(a^2-b^2-c^2)*(b^2-c^2)*x^3-(b^2-c^2)*(a^2-2*b^2-2*c^2)*x*y*z+a^4*(y-z)*y*z)] =0 (barys)

 

Q(X(2)) = X(2)X(11147) ∩ X(187)X(11258)

= (SB+SC)*(27*S^4-9*(3*(6*SA+SW)*R^2-3*SA^2-6*SB*SC+SW^2)*S^2+(6*SA+SW)*SW^3) : : (barys)

= on lines: {2, 11147}, {187, 11258}, {2930, 8586}, {5210, 14262}

=  [ -0.5922397161685296, -5.4938433559286030, 7.7174359050127330 ]

 

Q(X(3)) = ISOGONAL CONJUGATE OF X(22751)

= (5*R^2+SA-2*SW)*S^2-5*(3*R^2-SW)*SB*SC : : (barys)

= on lines: {3, 12278}, {4, 49}, {5, 13367}, {26, 12293}, {30, 511}, {68, 15138}, {110, 18403}, {125, 15646}, {143, 3575}, {186, 265}, {343, 550}, {382, 1993}, {403, 10113}, {546, 13403}, {568, 18559}, {1147, 18377}, {1495, 11563}, {1511, 2072}, {1568, 18572}, {1658, 9927}, {2070, 12902}, {2071, 12121}, {3153, 12383}, {3448, 13619}, {3618, 18420}, {3763, 7514}, {3853, 12897}, {5448, 18567}, {5449, 15331}, {5480, 19155}, {5576, 22804}, {5609, 18323}, {5654, 18568}, {5876, 14516}, {5899, 12412}, {5944, 10024}, {5946, 12022}, {6101, 12225}, {6102, 6240}, {6146, 13630}, {6241, 18565}, {6644, 18396}, {7564, 11425}, {7574, 15132}, {7577, 18430}, {7689, 18356}, {10095, 12241}, {10149, 12896}, {10151, 12140}, {10224, 12038}, {10226, 20299}, {10254, 11464}, {10255, 11449}, {10264, 21663}, {10282, 13406}, {10296, 23236}, {10610, 13160}, {10733, 14157}, {11250, 18381}, {11459, 18564}, {11591, 12605}, {12106, 18390}, {12111, 18562}, {12112, 12419}, {12118, 15139}, {12161, 12173}, {13142, 16982}, {13399, 16111}, {13490, 16657}, {14852, 18324}, {15114, 15122}, {18474, 18570}, {18945, 18952}, {19205, 19211}

= isogonal conjugate of X(22751)

= [ 1.3542662377745510, 0.9670591941770164, -1..2945484749415730 ]

 

Q(X(4)) = X(186)

 

Q( X(671) ) = X(1153)X(3054) ∩ X(8588)X(14650)

= (SB+SC)*(567*S^6-27*(6*(18*SA-SW)*R^2-21*SA^2+30*SB*SC+SW^2)*S^4-3*(9*SA^2+18*SB*SC+2*(9*R^2-5*SW)*SW)*SW^2*S^2+2*SW^6) : : (barys)

= on lines: {1153, 3054}, {8588, 14650}

= reflection of X(11841) in the line X(28585)X(29235)

= [ -2.4440992407121570, 1.7500389677929020, 3.5571448460718260 ]

 

Q( X(1113) ) = CIRCUMCIRCLE INVERSE OF X(28447)

= 3*(S^2-3*SB*SC)*a*b*c - 4*(SB+SC)*OH*SA*S : : (barys)

= 3*X(2100)+X(7982), 3*X(2104)-5*X(11482), X(11477)+3*X(15162)

= As a point on the Euler line, this center has Shinagawa coefficients [-3*R+2*OH, 9*R-2*OH]  

= on lines: {2, 3}, {2100, 7982}, {2104, 11482}, {2574, 5609}, {11477, 15162}

= midpoint of X(i) and X(j) for these {i,j}: {3, 15157}, {10751, 15160}

= reflection of X(20409) in X(140)

= circumperp conjugate of X(28448)

= circumcircle-inverse-of X(28447)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 15154, 15157), (1113, 1114, 28447), (1113, 15157, 3)

 

Q( X(1114) ) = CIRCUMCIRCLE INVERSE OF X(28448)

= 3*(S^2-3*SB*SC)*a*b*c + 4*(SB+SC)*OH*SA*S : : (barys)

= As a point on the Euler line, this center has Shinagawa coefficients [-3*R-2*OH, 9*R+2*OH]  

= on lines: {2, 3}, {2101, 7982}, {2105, 11482}, {2575, 5609}, {11477, 15163}

= midpoint of X(i) and X(j) for these {i,j}: {3, 15156}, {10750, 15161}

= reflection of X(20408) in X(140)

= circumperp conjugate of X(28447)

= circumcircle-inverse-of X(28448)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 15155, 15156), (140, 20408, 13626), (1113, 1114, 28448)

= [ 16.3692942528002100, 15.4565917183444100, -14.6151117474695400 ]

 

César Lozada