Let ABC be a triangle and P a point.
Denote:
Oa, Ob, Oc = the cirumcenters of IBC, ICA, IAB, resp.
Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
Which are the loci of P such that the NPCs of
1. IPOa, IPOb, IPOc
2. IPNa, IPNb, IPNc
are coaxial ? The entire plane?
[César Lozada]:
1) The entire plane.
The locus of Q1(P) (2nd point of intersection) is the circle {center=X(1385)=midpoint of {I,O}, radius=R/2}.
Q1(P) is a fixed point for all P (except I) on the conic q1={Oa, Ob, Oc, I, P} and Q1(P)=center-of-q1.
For P=u:v:w (trilinears)
Q1(P) = (b+c)^2*(2*a+b+c)*a*u^4+((b+c) *a*(2*a^2-(3*b-c)*a-(b+c)^2)* v+(b+c)*a*(2*a^2+(b-3*c)*a-(b+ c)^2)*w)*u^3+((-2*(2*b+c)*a^3+ (b^2-5*b*c-3*c^2)*a^2-b^2*(b+ c)*a-(b^2-c^2)*(b+c)*c)*v^2+( 2*a^4-(b+c)*a^3+6*a^2*b*c+2*( b+c)*(b^2+b*c+c^2)*a-(b^2-c^2) ^2)*w*v+(-2*(b+2*c)*a^3-(3*b^ 2+5*b*c-c^2)*a^2-c^2*(b+c)*a+( b^2-c^2)*(b+c)*b)*w^2)*u^2+(b* (a+c)*(2*a^2-(b-c)*a+b^2-c^2)* v^3+(-2*a^4+(b-c)*a^3-c*(b-3* c)*a^2-(b-c)*(2*b^2+c^2)*a+(b- c)*(b^3+c^3))*w*v^2+(-2*a^4-( b-c)*a^3+b*(3*b-c)*a^2+(b-c)*( b^2+2*c^2)*a-(b-c)*(b^3+c^3))* w^2*v+c*(a+b)*(2*a^2+(b-c)*a+ c^2-b^2)*w^3)*u+(a+c)^2*b^2*v^ 4-b*(a+c)*(2*a^2+(b+c)*a-(b-c) ^2)*w*v^3+(2*a^4+3*(b+c)*a^3+ 6*a^2*b*c-(b^2-c^2)*(b-c)*a-2* (b^2-b*c+c^2)*b*c)*w^2*v^2-c*( a+b)*(2*a^2+(b+c)*a-(b-c)^2)* w^3*v+(a+b)^2*c^2*w^4 : :
ETC pairs (P,Q1(P)):
(3,214), (21,214), (56,11719), (58,11700), (74,11709), (98,11710), (99,11711), (100,214), (101,11712), (102,11713), (103,11714), (104,11715), (105,11716), (106,11717), (107,11718), (108,11719), (109,11700), (110,11720), (111,11721), (112,11722), (224,214), (551,11711), (946,11718), (990,11719), (991,11700), (993,11711), (995,11712), (1001,214), (1064,11722), (1201,11720), (1385,11720), (1386,11720), (1394,11700), (2360,11712), (2574,11720), (2575,11720), (3307,214), (3308,214), (3413,11711), (3414,11711), (3576,11712), (3736,214), (3913,214), (4297,11711), (4653,11712), (5732,214), (6261,214), (6265,214), (7290,11700), (8283,11719), (8885,11719), (10571,11700), (11699,11720)
ETC pairs (Q1(P),P):
(214,3), (214,21), (214,100), (214,224), (214,1001), (214,3307), (214,3308), (214,3736), (214,3913), (214,5732), (214,6261), (214,6265), (11700,58), (11700,109), (11700,991), (11700,1394), (11700,7290), (11700,10571), (11709,74), (11710,98), (11711,99), (11711,551), (11711,993), (11711,3413), (11711,3414), (11711,4297), (11712,101), (11712,995), (11712,2360), (11712,3576), (11712,4653), (11713,102), (11714,103), (11715,104), (11716,105), (11717,106), (11718,107), (11718,946), (11719,56), (11719,108), (11719,990), (11719,8283), (11719,8885), (11720,110), (11720,1201), (11720,1385), (11720,1386), (11720,2574), (11720,2575), (11720,11699), (11721,111), (11722,112), (11722,1064)
Examples:
Q1(G) = Q1(9059) =
= 2*a^7-(b+c)*a^6-(b^2-8*b*c+c^ 2)*a^5+(b+c)*(5*b^2-13*b*c+5* c^2)*a^4-11*b*c*(b-c)^2*a^3-( b^2-c^2)*(b-c)*(2*b^2-3*b*c+2* c^2)*a^2+(b^2+c^2)*(b^2-3*b*c+ c^2)*(b+c)^2*a+b^2*c^2*(b+c)^3 : : (barycentrics)
= On lines: {1,9059}, {551,11717}
= midpoint of X(1) and X(9059)
= [ 2.964064245469369, 0.12733878966450, 2.184477206538477 ]
Q1(X(20)) = Q1(X(1305)) =
= 2*a^11-(b+c)*a^10-5*(b^2+c^2)* a^9+(b+c)*(3*b^2-b*c+3*c^2)*a^ 8+(4*b^4+4*c^4+b*c*(b^2+6*b*c+ c^2))*a^7-(b+c)*(5*b^4+5*c^4- b*c*(5*b^2-8*b*c+5*c^2))*a^6- b*c*(3*b^2-4*b*c+3*c^2)*(b+c)^ 2*a^5+(b+c)*(5*b^6+5*c^6-(7*b^ 4+7*c^4-4*b*c*(b^2+c^2))*b*c)* a^4-(b^2-c^2)^2*(b-c)^2*(2*b^ 2+b*c+2*c^2)*a^3-(b^4-c^4)*(b^ 2-c^2)*(b+c)*(2*b^2-3*b*c+2*c^ 2)*a^2+(b^2-c^2)*(b-c)*(b^3-c^ 3)*(b^4-c^4)*a+(b^2-c^2)^3*(b- c)*b^2*c^2 : : (barycentrics)
= On lines: {1,1305}, {917,3576}, {1125,5190}, {4297,11714}
= midpoint of X(1) and X(1305)
= reflection of X(5190) in X(1125)
= [ 0.548079525417553, 2.35270158539802, 1.758911295669958 ]
2) The entire plane.
The locus of Q2(P) (2nd point of intersection) is the circle {center=X(5901), radius=sqrt(R^2-2*R*r)/4}.
Q2(P) is a fixed point for all P (except I) on the conic q2={Na, Nb, Nc, I, P} and Q2(P)=center-of-q2.
For P=u:v:w (trilinears), Q2(P) has very long coordinates (given below).
ETC-pairs (P,Q2(P)): (5,1125), (10,1125), (21,1125), (1385,1125), (3307,1125), (3308,1125), (5450,1125), (5506,1125), (7705,1125), (10129,1125), (11263,1125)
ETC-pairs (Q2(P), P): (1125,5), (1125,10), (1125,21), (1125,1385), (1125,3307), (1125,3308), (1125,5450), (1125,5506), (1125,7705), (1125,10129), (1125,11263)
César Lozada
Q2(P) = (a^3*(2*a^8+(b+c)*a^7-2*(3*b^ 2+4*b*c+3*c^2)*a^6-2*(b+c)*(2* b^2-3*b*c+2*c^2)*a^5+(5*b^4+5* c^4+8*b*c*(2*b^2+3*b*c+2*c^2)) *a^4+(b+c)*(5*b^4+5*c^4-b*c*( 7*b^2+11*b*c+7*c^2))*a^3-2*b* c*(5*b^2-4*b*c+5*c^2)*(b+c)^2* a^2-(2*b^4+2*c^4-b*c*(5*b^2-2* b*c+5*c^2))*(b+c)^3*a+(b^2-c^ 2)^2*(b+c)^2*(-c^2+4*b*c-b^2)) *u^4+(-a^2*(2*a^9+2*a^8*b-(b^ 2+14*b*c+9*c^2)*a^7-(5*b^3+c^ 3-b*c*(13*b-11*c))*a^6-(11*b^ 4-15*c^4-2*b*c*(8*b^2+19*b*c+ 16*c^2))*a^5+(b^5+3*c^5-(13*b^ 3-21*c^3+2*b*c*(16*b+13*c))*b* c)*a^4+(b+c)*(17*b^5-11*c^5-( 21*b^3+7*c^3-b*c*(31*b-43*c))* b*c)*a^3+(5*b^5-3*c^5-(9*b^3+ 7*c^3-b*c*(b+25*c))*b*c)*(b+c) ^2*a^2-(b^2-c^2)*(b+c)^2*(7*b^ 4+3*c^4-2*b*c*(8*b-3*c)*(b-c)) *a+(b^2-c^2)^2*(b+c)^2*(-3*b^ 3+c^3+b*c*(5*b-c)))*v-a^2*(2* a^9+2*a^8*c-(9*b^2+14*b*c+c^2) *a^7-(b^3+5*c^3+b*c*(11*b-13* c))*a^6+(15*b^4-11*c^4+2*b*c*( 16*b^2+19*b*c+8*c^2))*a^5+(3* b^5+c^5+(21*b^3-13*c^3-2*b*c*( 13*b+16*c))*b*c)*a^4-(b+c)*( 11*b^5-17*c^5+(7*b^3+21*c^3+b* c*(43*b-31*c))*b*c)*a^3-(3*b^ 5-5*c^5+(7*b^3+9*c^3-b*c*(25* b+c))*b*c)*(b+c)^2*a^2+(b^2-c^ 2)*(b+c)^2*(3*b^4+7*c^4-2*b*c* (b-c)*(3*b-8*c))*a+(b^2-c^2)^ 2*(b+c)^2*(b^3-3*c^3-b*c*(b-5* c)))*w)*u^3+(a*b*(a^9+(5*b-4* c)*a^8+(2*b^2+9*b*c-7*c^2)*a^ 7-2*(7*b^3-7*c^3+b*c*(3*b-8*c) )*a^6-(12*b^4-19*c^4-2*b*c*(2* b^2-24*b*c+3*c^2))*a^5+(12*b^ 5-14*c^5-(2*b^3+31*c^3-4*b*c*( 17*b-7*c))*b*c)*a^4+(b+c)*(14* b^5-21*c^5-(31*b^3+7*c^3+b*c*( 9*b-59*c))*b*c)*a^3-(b^2-c^2)* (b+c)*(2*b^4+2*c^4-b*c*(16*b^ 2-37*b*c-c^2))*a^2-(b^2-c^2)*( b+c)^2*(5*b^4+8*c^4-b*c*(14*b^ 2-6*b*c+7*c^2))*a+(b^2-c^2)^3* (b+c)*(-b^2-b*c-2*c^2))*v^2+a* (2*a^10-(9*b^2+20*b*c+9*c^2)* a^8-(b+c)*(b^2+4*b*c+c^2)*a^7+ 3*(5*b^4+5*c^4+8*b*c*(b^2+3*b* c+c^2))*a^6+(b+c)*(3*b^4-26*b^ 2*c^2+3*c^4)*a^5-(11*b^6+11*c^ 6-2*(7*b^4+7*c^4-4*b*c*(9*b^2+ b*c+9*c^2))*b*c)*a^4-(b+c)*(3* b^6+3*c^6-2*(8*b^4+8*c^4+b*c*( 9*b^2-10*b*c+9*c^2))*b*c)*a^3+ (b^2-c^2)^2*(3*b^4+3*c^4-2*b* c*(8*b^2-9*b*c+8*c^2))*a^2+(b^ 2-c^2)^2*(b+c)*(b^4+c^4-2*b*c* (6*b^2-7*b*c+6*c^2))*a-2*(b^2- c^2)^4*b*c)*w*v+a*c*(a^9-(4*b- 5*c)*a^8-(7*b^2-9*b*c-2*c^2)* a^7+2*(7*b^3-7*c^3+b*c*(8*b-3* c))*a^6+(19*b^4-12*c^4+2*b*c*( 3*b^2-24*b*c+2*c^2))*a^5-(14* b^5-12*c^5+(31*b^3+2*c^3+4*b* c*(7*b-17*c))*b*c)*a^4-(b+c)*( 21*b^5-14*c^5+(7*b^3+31*c^3-b* c*(59*b-9*c))*b*c)*a^3+(b^2-c^ 2)*(b+c)*(2*b^4+2*c^4+b*c*(b^ 2+37*b*c-16*c^2))*a^2+(b^2-c^ 2)*(b+c)^2*(8*b^4+5*c^4-b*c*( 7*b^2-6*b*c+14*c^2))*a+(b^2-c^ 2)^3*(b+c)*(2*b^2+b*c+c^2))*w^ 2)*u^2+(-b^2*(4*a^8*b+(b+3*c)* (b+c)*a^7-(11*b^3-5*c^3-5*b*c* (3*b-5*c))*a^6-(b^4+5*c^4+2*b* c*(14*b^2-33*b*c+20*c^2))*a^5+ (11*b^5-13*c^5-(15*b^3+c^3+6* b*c*(7*b-12*c))*b*c)*a^4-(b^2- c^2)*(b^4-3*c^4-b*c*(30*b^2- 15*b*c-22*c^2))*a^3-(b^2-c^2)* (b+c)*(5*b^4+7*c^4-b*c*(4*b^2+ 13*b*c-2*c^2))*a^2+(b^2-c^2)^ 2*(b+c)^2*(b^2-8*b*c+5*c^2)*a+ (b^2-c^2)^4*(b+c))*v^3+b*(2*( b+c)*a^9+4*b*c*a^8-(9*b^3+6*c^ 3-b*c*(26*b-35*c))*a^7-(b^3+ 43*c^3+b*c*(5*b-57*c))*b*a^6+( 15*b^5+6*c^5-(56*b^3-27*c^3-4* b*c*(8*b-c))*b*c)*a^5+(3*b^5+ 35*c^5+(3*b^3-61*c^3-2*b*c*( 31*b-35*c))*b*c)*b*a^4-(b^2-c^ 2)*(11*b^5-2*c^5-(30*b^3-3*c^ 3+b*c*(5*b+22*c))*b*c)*a^3-3*( b^2-c^2)*(b+c)*b*(b^4-b^2*c^2+ c^4)*a^2+(b^2-c^2)^2*(b+c)^2* b*(3*b^2-8*b*c+3*c^2)*a+(b^2- c^2)^4*(b+c)*b)*w*v^2+c*(2*(b+ c)*a^9+4*b*c*a^8-(6*b^3+9*c^3+ (35*b-26*c)*b*c)*a^7-(43*b^3+ c^3-(57*b-5*c)*b*c)*c*a^6+(6* b^5+15*c^5+(27*b^3-56*c^3-4*( b-8*c)*b*c)*b*c)*a^5+(35*b^5+ 3*c^5-(61*b^3-3*c^3-2*b*c*(35* b-31*c))*b*c)*c*a^4-(b^2-c^2)* (2*b^5-11*c^5-(3*b^3-30*c^3-b* c*(22*b+5*c))*b*c)*a^3+3*(b^2- c^2)*(b+c)*c*(b^4-b^2*c^2+c^4) *a^2+(b^2-c^2)^2*(b+c)^2*c*(3* b^2-8*b*c+3*c^2)*a+(b^2-c^2)^ 4*(b+c)*c)*w^2*v-c^2*(4*a^8*c+ (b+c)*(3*b+c)*a^7+(5*b^3-11*c^ 3-5*b*c*(5*b-3*c))*a^6-(5*b^4+ c^4+2*b*c*(20*b^2-33*b*c+14*c^ 2))*a^5-(13*b^5-11*c^5+(b^3+ 15*c^3-6*b*c*(12*b-7*c))*b*c)* a^4-(b^2-c^2)*(3*b^4-c^4-b*c*( 22*b^2+15*b*c-30*c^2))*a^3+(b^ 2-c^2)*(b+c)*(7*b^4+5*c^4+b*c* (2*b^2-13*b*c-4*c^2))*a^2+(b^ 2-c^2)^2*(b+c)^2*(5*b^2-8*b*c+ c^2)*a+(b^2-c^2)^4*(b+c))*w^3) *u+b^3*(a^8-(b-4*c)*a^7-(3*b^ 2-c^2)*a^6+2*(2*b-5*c)*(b^2-b* c+c^2)*a^5+(4*b^4-9*c^4+2*b*c* (3*b^2-13*b*c+13*c^2))*a^4-(5* b^5-4*c^5-2*(7*b^3+7*c^3+3*b* c*(b-5*c))*b*c)*a^3-(b^2-c^2)* (3*b^4+7*c^4+b*c*(7*b^2-15*b* c-c^2))*a^2+(b^2-c^2)^2*(2*b^ 3+2*c^3-b*c*(4*b+3*c))*a+(b^2- c^2)^3*(b+c)*b)*v^4-b^2*(2*a^ 9+6*a^8*c-(3*b-c)^2*a^7-(b^3+ 17*c^3+b*c*(23*b-29*c))*a^6+( 15*b^4-11*c^4-2*b*c*(6*b^2+9* b*c-14*c^2))*a^5+(3*b^5+11*c^ 5+(33*b^3-15*c^3-2*b*c*(24*b- 11*c))*b*c)*a^4-(b-c)*(11*b^5+ 13*c^5+(b^3-11*c^3-5*b*c*(5*b- 3*c))*b*c)*a^3-(b^2-c^2)*(3*b^ 5+c^5+(17*b^3-c^3-7*b*c*(3*b- c))*b*c)*a^2+(b^2-c^2)^2*(b-c) *(3*b^3+3*c^3-b*c*(b-c))*a+(b^ 4-c^4)*(b^2-c^2)^2*(b+c))*w*v^ 3+b*c*a*(4*a^8+5*(b+c)*a^7-2*( 7*b^2-11*b*c+7*c^2)*a^6-(b+c)* (19*b^2-24*b*c+19*c^2)*a^5+2*( 7*b^2-10*b*c+7*c^2)*(b^2-b*c+ c^2)*a^4+(b+c)*(23*b^4+23*c^4- b*c*(39*b^2-41*b*c+39*c^2))*a^ 3-(2*b^6+2*c^6-(13*b^4+13*c^4- 4*b*c*(10*b^2-13*b*c+10*c^2))* b*c)*a^2-(b^2-c^2)*(b-c)*(9*b^ 4+9*c^4-b*c*(b^2-13*b*c+c^2))* a-(b^2-c^2)^2*(2*b^4+2*c^4-b* c*(3*b^2+4*b*c+3*c^2)))*w^2*v^ 2-c^2*(2*a^9+6*a^8*b-(b-3*c)^ 2*a^7-(17*b^3+c^3-b*c*(29*b- 23*c))*a^6-(11*b^4-15*c^4-2*b* c*(14*b^2-9*b*c-6*c^2))*a^5+( 11*b^5+3*c^5-(15*b^3-33*c^3-2* b*c*(11*b-24*c))*b*c)*a^4+(b- c)*(13*b^5+11*c^5-(11*b^3-c^3- 5*b*c*(3*b-5*c))*b*c)*a^3+(b^ 2-c^2)*(b^5+3*c^5-(b^3-17*c^3- 7*b*c*(b-3*c))*b*c)*a^2-(b^2- c^2)^2*(b-c)*(3*b^3+3*c^3+b*c* (b-c))*a+(b^4-c^4)*(b^2-c^2)^ 2*(-b-c))*w^3*v+c^3*(a^8+(4*b- c)*a^7+(b^2-3*c^2)*a^6-2*(5*b- 2*c)*(b^2-b*c+c^2)*a^5-(9*b^4- 4*c^4-2*b*c*(13*b^2-13*b*c+3* c^2))*a^4+(4*b^5-5*c^5+2*(7*b^ 3+7*c^3-3*b*c*(5*b-c))*b*c)*a^ 3+(b^2-c^2)*(7*b^4+3*c^4-b*c*( b^2+15*b*c-7*c^2))*a^2+(b^2-c^ 2)^2*(2*b^3+2*c^3-b*c*(3*b+4* c))*a-(b^2-c^2)^3*(b+c)*c)*w^ 4)/a
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