HYACINTHOS 29101

[Antreas P. Hatzipolakis]:

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote Na, Nb, Nc = the NPC centers pf PBC, PCA, PAB, resp.

O = the circumcenter of NaNbNc

Oa, Ob, Oc = the circumcenters of O'NbNc, O'NcNa, O'NaNb, resp.

O1, O2, O3 = the reflections of Oa, Ob, Oc in PA', PB', PC', resp.

Which is the locus of P such that ABC, O1O2O3 are orthologic?
(I lies on the locus. The orthologic center (ABC, O1O2O3) lies on the circumcircle)


[César Lozada]:

 

 

Locus= {sidelines} ∪ {Linf}  ∪ {circumcircle} ∪ {Q030, through ETCs 1, 4, 13, 14, 74, 80}

 

Orthologic centers:

·         For P ∈ Q030

A->O1 = Qa( X(1) ) = X(953)

O1->A = Qo( X(1) ) = MIDPOINT OF X(145) AND X(6796)

= 4*a^7-8*(b+c)*a^6-(3*b^2-22*b*c+3*c^2)*a^5+(b+c)*(15*b^2-31*b*c+15*c^2)*a^4-(6*b^2+25*b*c+6*c^2)*(b-c)^2*a^3-(b^2-c^2)*(b-c)*(6*b^2-17*b*c+6*c^2)*a^2+(b^2-c^2)^2*(5*b^2-9*b*c+5*c^2)*a-(b^2-c^2)^3*(b-c) : : (barys)

= X(5450)-3*X(7967), X(6705)-3*X(13607)

= lies on these lines: {1, 6830}, {145, 6796}, {515, 1483}, {952, 24387}, {1071, 1317}, {3244, 24474}, {3746, 5450}, {3825, 19907}, {6705, 13607}, {7972, 21740}, {7982, 11015}, {10572, 25485}, {12608, 21630}

= midpoint of X(145) and X(6796)

= reflection of X(24387) in X(26087)

= [ 10.9083443399498900, 12.7166516430966200, -10.1977917355978500 ]

 

Qa( X(4) ) = X(74)

Qo( X(4) ) = X(3627)

 

Qa( X(13) ) = ISOGONAL CONJUGATE OF X(22998)

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

= lies on these lines: {187, 6105}, {623, 11133}, {10677, 32301}

= isogonal conjugate of X(22998)

= [ -3.3597044918699330, 3.8070509637566160, 2.5556466570928440 ]

 

Qo( X(13) ) = REFLECTION OF X(2) IN X(20415)

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

= 3*SB*SC*SW+(9*SA-8*SW)*S^2-2*sqrt(3)*S*(S^2+3*SB*SC) : : (barys)

= 3*X(13)-X(381), 4*X(13)-X(22796), 4*X(381)-3*X(22796), 2*X(547)-3*X(5459), 2*X(549)-3*X(6771), 3*X(616)-7*X(15702), 3*X(618)-4*X(10124), X(3543)+3*X(6770), X(3543)-3*X(25154), 5*X(5071)-3*X(5617), 3*X(5463)-5*X(15694), 3*X(5473)-5*X(14093), 3*X(5478)-2*X(14893), 3*X(9166)-X(22507), 2*X(11737)-3*X(20252), 3*X(13103)+X(15681), 7*X(15700)-9*X(21156), 7*X(15703)-9*X(22489)

= lies on these lines: {2, 20415}, {6, 13}, {30, 16001}, {182, 22492}, {376, 9735}, {530, 549}, {543, 25559}, {547, 5459}, {575, 31694}, {616, 15702}, {618, 10124}, {3543, 6770}, {5071, 5617}, {5463, 15694}, {5473, 14093}, {5478, 14893}, {6036, 9762}, {9166, 22507}, {11302, 12155}, {11737, 20252}, {12243, 22509}, {13103, 15681}, {15700, 21156}, {15703, 22489}, {19924, 20425}

= midpoint of X(i) and X(j) for these {i,j}: {6770, 25154}, {6778, 11632}, {12243, 22509}

= reflection of X(2) in X(20415)

= [ 0.8757303823163774, 1.3087675226259020, 2.3304114051742720 ]

 

Qa( X(14) ) = ISOGONAL CONJUGATE OF X(22997)

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

= lies on these lines: {187, 6104}, {624, 11132}, {10678, 32302}

= isogonal conjugate of X(22997)

= [ 7.0570126129558370, 2.3899901976774090, -1.2710268609257530 ]

 

Qo( X(14) ) = REFLECTION OF X(2) IN X(20416)

= 2*(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2)*sqrt(3)*S+5*a^6-8*(b^2+c^2)*a^4+2*(2*b^4-3*b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2)

= 3*SB*SC*SW+(9*SA-8*SW)*S^2+2*sqrt(3)*S*(S^2+3*SB*SC) : : (barys)

= 3*X(14)-X(381), 4*X(14)-X(22797), 4*X(381)-3*X(22797), 2*X(547)-3*X(5460), 2*X(549)-3*X(6774), 3*X(617)-7*X(15702), 3*X(619)-4*X(10124), X(3543)+3*X(6773), X(3543)-3*X(25164), 5*X(5071)-3*X(5613), 3*X(5464)-5*X(15694), 3*X(5474)-5*X(14093), 3*X(5479)-2*X(14893), 3*X(9166)-X(22509), 2*X(11737)-3*X(20253), 3*X(13102)+X(15681), 7*X(15700)-9*X(21157), 7*X(15703)-9*X(22490)

= lies on these lines: {2, 20416}, {6, 13}, {30, 16002}, {182, 22491}, {376, 9736}, {531, 549}, {543, 25560}, {547, 5460}, {575, 31693}, {617, 15702}, {619, 10124}, {3543, 6773}, {5071, 5613}, {5464, 15694}, {5474, 14093}, {5479, 14893}, {6036, 9760}, {9166, 22509}, {11301, 12154}, {11737, 20253}, {12243, 22507}, {13102, 15681}, {15700, 21157}, {15703, 22490}, {19924, 20426}

= midpoint of X(i) and X(j) for these {i,j}: {6773, 25164}, {6777, 11632}, {12243, 22507}

= reflection of X(2) in X(20416)

= [ 7.3219603126295960, 11.2437142882360800, -7.5228117080850300 ]

 

Qa( X(74) ) = Not interesting

Qo( X(74) ) = X(10990)

 

Qa( X(80) ) = Not interesting

Qo( X(80) ) = X(80)X(5450) ∩ X(2800)X(12688)

= 4*a^10-11*(b+c)*a^9+(b^2+42*b*c+c^2)*a^8+6*(b+c)*(4*b^2-11*b*c+4*c^2)*a^7-(24*b^4+24*c^4+(29*b^2-114*b*c+29*c^2)*b*c)*a^6-(b+c)*(6*b^4+6*c^4-(95*b^2-177*b*c+95*c^2)*b*c)*a^5+(26*b^4+26*c^4-(10*b^2+99*b*c+10*c^2)*b*c)*(b-c)^2*a^4-(b^2-c^2)*(b-c)*(16*b^4+16*c^4+(24*b^2-83*b*c+24*c^2)*b*c)*a^3-(4*b^4+4*c^4-(43*b^2-79*b*c+43*c^2)*b*c)*(b^2-c^2)^2*a^2+(b^2-c^2)^3*(b-c)*(9*b^2-19*b*c+9*c^2)*a-3*(b^2-c^2)^4*(b-c)^2 : : (barys)

= 3*X(80)-X(5450), X(11500)+3*X(12747)

= lies on these lines: {80, 5450}, {2800, 12688}, {11500, 12747}

= [ 10.0833602151226200, 13.6532413617711200, -10.4654380986060700 ]

 

·         For P ∈ circumcircle-of-ABC

Some ETC-pairs (P,Qo(P)): (74,10990), (98, 10991), (99, 10992), (100, 10993)

 

Notes:

-          The locus of Qo(P) when P moves on the circumcircle of ABC is the circle with center X(550) and radius 3*R/2.

-          This configuration seems to be related to Hyacinthos 24476

 

César Lozada