Let ABC be a triangle and P a point.
Denote:
Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.
Oa, Ob, Oc = the circumcenters of PBC, PCA, PAB, resp.
Ha, Hb, Hc = the orthocenters of PBC, PCA, PAB, resp.
Which are the loci of P such that the triangles NaNbNc, OaObOc, HaHbHc in pairs are orthologic or parallelogic?
That is:
1. NaNbNc, OaObOc are orthologic
2. NaNbNc, OaObOc are parallelogic
3. NaNbNc, HaHbHc are orthologic
4. NaNbNc, HaHbHc are parallelogic
5. OaObOc, HaHbHc are orthologic
6. OaObOc, HaHbHc are parallelogic
[CÉSAR LOZADA]
Loci for orthologic triangles = The entire plane for all pairs
Loci for parallelogic triangles =
1) Locus = The entire plane for NaNbNc, OaObOc
2) Locus = {sidelines} ∪ {Linf} ∪ {circumcircle}, for OaObOc, HaHbHc
3) Locus = {sidelines} ∪ {Linf} ∪ {quartic ∑[ y*z*((SB+SC)^2*y*z-2*(S^2-SA^2)*x^2)]=0, through ETCs X(4)}, for NaNbNc, HaHbHc
Assume P=x:y:z (barys)
Orthologic centers:
Qon(P) = (Oaà Na) = (-(-a^2+b^2+c^2)*c^6*x^4*y^4+(-a^2+b^2+c^2)*(a^2-3*b^2-c^2)*c^4*x^4*y^3*z-(-a^2+b^2+c^2)*(a^6-2*a^4*b^2-2*a^4*c^2+a^2*b^4-4*a^2*b^2*c^2+a^2*c^4+3*b^4*c^2+3*b^2*c^4)*x^4*y^2*z^2+(a^2-b^2-3*c^2)*(-a^2+b^2+c^2)*b^4*x^4*y*z^3-(-a^2+b^2+c^2)*b^6*x^4*z^4+(a^2-b^2+c^2)*c^6*x^3*y^5+(2*a^2-2*b^2-c^2)*(a^2+b^2-c^2)*c^4*x^3*y^4*z+(3*a^8-9*a^6*b^2-11*a^6*c^2+9*a^4*b^4+12*a^4*b^2*c^2+13*a^4*c^4-3*a^2*b^6-6*a^2*b^2*c^4-5*a^2*c^6-b^6*c^2+b^2*c^6)*x^3*y^3*z^2+(3*a^8-11*a^6*b^2-9*a^6*c^2+13*a^4*b^4+12*a^4*b^2*c^2+9*a^4*c^4-5*a^2*b^6-6*a^2*b^4*c^2-3*a^2*c^6+b^6*c^2-b^2*c^6)*x^3*y^2*z^3+(2*a^2-b^2-2*c^2)*(a^2-b^2+c^2)*b^4*x^3*y*z^4+(a^2+b^2-c^2)*b^6*x^3*z^5+(3*a^4-4*a^2*b^2+2*a^2*c^2+b^4-2*b^2*c^2+c^4)*c^4*x^2*y^5*z+(3*a^8-9*a^6*b^2-8*a^6*c^2+9*a^4*b^4+9*a^4*b^2*c^2+6*a^4*c^4-3*a^2*b^6-3*a^2*b^4*c^2+a^2*b^2*c^4-3*a^2*c^6+2*b^6*c^2-2*b^4*c^4-2*b^2*c^6+2*c^8)*x^2*y^4*z^2+(5*a^8-16*a^6*b^2-16*a^6*c^2+18*a^4*b^4+6*a^4*b^2*c^2+18*a^4*c^4-8*a^2*b^6+2*a^2*b^4*c^2+2*a^2*b^2*c^4-8*a^2*c^6+b^8+2*b^6*c^2-6*b^4*c^4+2*b^2*c^6+c^8)*x^2*y^3*z^3+(3*a^8-8*a^6*b^2-9*a^6*c^2+6*a^4*b^4+9*a^4*b^2*c^2+9*a^4*c^4-3*a^2*b^6+a^2*b^4*c^2-3*a^2*b^2*c^4-3*a^2*c^6+2*b^8-2*b^6*c^2-2*b^4*c^4+2*b^2*c^6)*x^2*y^2*z^4+(3*a^4+2*a^2*b^2-4*a^2*c^2+b^4-2*b^2*c^2+c^4)*b^4*x^2*y*z^5+(a^6-3*a^4*b^2+3*a^2*b^4-3*a^2*b^2*c^2+4*a^2*c^4-b^6+3*b^4*c^2-3*b^2*c^4+c^6)*a^2*x*y^5*z^2+(3*a^6-7*a^4*b^2-10*a^4*c^2+5*a^2*b^4+a^2*b^2*c^2+6*a^2*c^4-b^6+3*b^4*c^2-3*b^2*c^4+c^6)*a^2*x*y^4*z^3+(3*a^6-10*a^4*b^2-7*a^4*c^2+6*a^2*b^4+a^2*b^2*c^2+5*a^2*c^4+b^6-3*b^4*c^2+3*b^2*c^4-c^6)*a^2*x*y^3*z^4+(a^6-3*a^4*c^2+4*a^2*b^4-3*a^2*b^2*c^2+3*a^2*c^4+b^6-3*b^4*c^2+3*b^2*c^4-c^6)*a^2*x*y^2*z^5+(a^4-2*a^2*b^2+b^4-2*b^2*c^2+c^4)*a^4*y^5*z^3-2*(a^2*b^2+a^2*c^2-b^4+2*b^2*c^2-c^4)*a^4*y^4*z^4+(a^4-2*a^2*c^2+b^4-2*b^2*c^2+c^4)*a^4*y^3*z^5)*x : :
Qon(P ∈ circumcircle) = X(3); Qon(P ∈ Linf) = P
Some other ETC pairs: (1,100), (2,16508), (3,12893), (6,16510), (80,8), (265,9927)
------
Qno(P) = (Naà Oa) = (-(-a^2+b^2+c^2)*c^2*x^2*y^2-4*b^2*c^2*x^2*y*z-(-a^2+b^2+c^2)*b^2*x^2*z^2+(a^2-b^2+c^2)*c^2*x*y^3+(a^4-2*a^2*b^2-3*a^2*c^2+b^4-3*b^2*c^2+2*c^4)*x*y^2*z+(a^4-3*a^2*b^2-2*a^2*c^2+2*b^4-3*b^2*c^2+c^4)*x*y*z^2+(a^2+b^2-c^2)*b^2*x*z^3+(a^4-2*a^2*b^2+b^4-2*b^2*c^2+c^4)*y^3*z-2*(a^2*b^2+a^2*c^2-b^4+2*b^2*c^2-c^4)*y^2*z^2+(a^4-2*a^2*c^2+b^4-2*b^2*c^2+c^4)*y*z^3)*x : :
Qno(P ∈ circumcircle) = X(5); Qno(P ∈ Linf) = P
Some other ETC pairs: (1,10), (2,16509), (3,5449), (4,5), (5,32551), (6,16511), (11,33528), (13,2), (14,2), (15,33529), (16,33530), (20,33531), (67,16511), (80,10), (265,5449), (671,16509)
====================
Qho(P) = (Haà Oa) = x/(-x*(SA*x-y*SB-z*SC)+(SB+SC)*y*z) : :
Qho(P ∈ circumcircle) = X(4); Qho(P ∈ Linf) = P
Some other ETC pairs: (1,8), (2,5485), (3,68), (4,4), (5,25043), (6,5486), (13,2), (14,2), (15,19778), (16,19779), (17,5487), (54,13418), (67,5486), (79,10266), (80,8), (84,10309), (115,23105), (125,5489), (186,562), (265,68), (485,5490), (502,6757), (671,5485)
--------
Qoh(P) = (Oaà Ha) = (-4*(-a^2+b^2+c^2)*c^6*x^4*y^4+4*(-a^2+b^2+c^2)*(a^2-3*b^2-c^2)*c^4*x^4*y^3*z+(a^4-a^2*b^2-a^2*c^2-12*b^2*c^2)*(-a^2+b^2+c^2)^2*x^4*y^2*z^2+4*(a^2-b^2-3*c^2)*(-a^2+b^2+c^2)*b^4*x^4*y*z^3-4*(-a^2+b^2+c^2)*b^6*x^4*z^4+4*(a^2-b^2+c^2)*c^6*x^3*y^5+2*(3*a^4+2*a^2*b^2-4*a^2*c^2-5*b^4+4*b^2*c^2+c^4)*c^4*x^3*y^4*z+(3*a^8-9*a^6*b^2-17*a^6*c^2+9*a^4*b^4+26*a^4*b^2*c^2+23*a^4*c^4-3*a^2*b^6-a^2*b^4*c^2-21*a^2*b^2*c^4-7*a^2*c^6-8*b^6*c^2+6*b^4*c^4+4*b^2*c^6-2*c^8)*x^3*y^3*z^2+(3*a^8-17*a^6*b^2-9*a^6*c^2+23*a^4*b^4+26*a^4*b^2*c^2+9*a^4*c^4-7*a^2*b^6-21*a^2*b^4*c^2-a^2*b^2*c^4-3*a^2*c^6-2*b^8+4*b^6*c^2+6*b^4*c^4-8*b^2*c^6)*x^3*y^2*z^3+2*(3*a^4-4*a^2*b^2+2*a^2*c^2+b^4+4*b^2*c^2-5*c^4)*b^4*x^3*y*z^4+4*(a^2+b^2-c^2)*b^6*x^3*z^5+2*(a^2-b^2+c^2)*(5*a^2-b^2+c^2)*c^4*x^2*y^5*z+(3*a^8-9*a^6*b^2-9*a^6*c^2+9*a^4*b^4+22*a^4*b^2*c^2+a^4*c^4-3*a^2*b^6-17*a^2*b^4*c^2+19*a^2*b^2*c^4+a^2*c^6+4*b^6*c^2-4*b^4*c^4-4*b^2*c^6+4*c^8)*x^2*y^4*z^2+2*(a^8+c^8-7*a^2*c^6-6*b^4*c^4-7*a^6*c^2+2*b^2*c^6-2*a^4*b^2*c^2+12*a^4*b^4+b^8+7*a^2*b^2*c^4+2*b^6*c^2+7*a^2*b^4*c^2-7*a^2*b^6+12*a^4*c^4-7*a^6*b^2)*x^2*y^3*z^3+(3*a^8-9*a^6*b^2-9*a^6*c^2+a^4*b^4+22*a^4*b^2*c^2+9*a^4*c^4+a^2*b^6+19*a^2*b^4*c^2-17*a^2*b^2*c^4-3*a^2*c^6+4*b^8-4*b^6*c^2-4*b^4*c^4+4*b^2*c^6)*x^2*y^2*z^4+2*(5*a^2+b^2-c^2)*(a^2+b^2-c^2)*b^4*x^2*y*z^5+(a^6-3*a^4*b^2+5*a^4*c^2+3*a^2*b^4-10*a^2*b^2*c^2+15*a^2*c^4-b^6+5*b^4*c^2-7*b^2*c^4+3*c^6)*a^2*x*y^5*z^2+(a^6-a^4*b^2-17*a^4*c^2-a^2*b^4+14*a^2*b^2*c^2+11*a^2*c^4+b^6+3*b^4*c^2-9*b^2*c^4+5*c^6)*a^2*x*y^4*z^3+(a^6-17*a^4*b^2-a^4*c^2+11*a^2*b^4+14*a^2*b^2*c^2-a^2*c^4+5*b^6-9*b^4*c^2+3*b^2*c^4+c^6)*a^2*x*y^3*z^4+(a^6+5*a^4*b^2-3*a^4*c^2+15*a^2*b^4-10*a^2*b^2*c^2+3*a^2*c^4+3*b^6-7*b^4*c^2+5*b^2*c^4-c^6)*a^2*x*y^2*z^5+2*(a^2-b^2+c^2)^2*a^4*y^5*z^3-4*(a^2+b^2-c^2)*(a^2-b^2+c^2)*a^4*y^4*z^4+2*(a^2+b^2-c^2)^2*a^4*y^3*z^5)*x : :
Qoh(P ∈ circumcircle) = X(3); Qoh(P ∈ Linf) = P
Some other ETC pairs: (1,3913), (3,9932), (4,3), (80,10912)
====================
Qnh(P) = (Naà Ha) = (4*(-a^2+b^2+c^2)*c^6*x^4*y^4-2*(a^2-b^2-3*c^2)*(-a^2+b^2+c^2)^2*c^2*x^4*y^3*z+(2*a^8-9*a^6*b^2-9*a^6*c^2+15*a^4*b^4+26*a^4*b^2*c^2+15*a^4*c^4-11*a^2*b^6-25*a^2*b^4*c^2-25*a^2*b^2*c^4-11*a^2*c^6+3*b^8+8*b^6*c^2+26*b^4*c^4+8*b^2*c^6+3*c^8)*x^4*y^2*z^2-2*(a^2-3*b^2-c^2)*(-a^2+b^2+c^2)^2*b^2*x^4*y*z^3+4*(-a^2+b^2+c^2)*b^6*x^4*z^4-4*(a^2-b^2+c^2)*c^6*x^3*y^5+2*(a^2-b^2+c^2)*(-a^2+b^2+c^2)*(2*a^2-2*b^2-c^2)*c^2*x^3*y^4*z+(3*a^8-15*a^6*b^2-7*a^6*c^2+27*a^4*b^4+10*a^4*b^2*c^2+13*a^4*c^4-21*a^2*b^6+a^2*b^4*c^2+21*a^2*b^2*c^4-17*a^2*c^6+6*b^8-4*b^6*c^2+6*b^4*c^4-16*b^2*c^6+8*c^8)*x^3*y^3*z^2+(3*a^8-7*a^6*b^2-15*a^6*c^2+13*a^4*b^4+10*a^4*b^2*c^2+27*a^4*c^4-17*a^2*b^6+21*a^2*b^4*c^2+a^2*b^2*c^4-21*a^2*c^6+8*b^8-16*b^6*c^2+6*b^4*c^4-4*b^2*c^6+6*c^8)*x^3*y^2*z^3+2*(-a^2+b^2+c^2)*(2*a^2-b^2-2*c^2)*(a^2+b^2-c^2)*b^2*x^3*y*z^4-4*(a^2+b^2-c^2)*b^6*x^3*z^5-2*(a^2-b^2+2*c^2)*(a^2-b^2+c^2)^2*c^2*x^2*y^5*z-(-17*a^2*b^4*c^2-a^2*b^2*c^4+5*c^8-10*b^4*c^4+7*a^6*c^2-3*b^8-2*a^4*b^2*c^2+12*b^6*c^2+9*a^2*b^6-4*b^2*c^6+3*a^6*b^2-9*a^4*b^4+9*a^2*c^6-21*a^4*c^4)*x^2*y^4*z^2+2*(16*a^4*b^2*c^2+9*a^4*b^4+18*b^4*c^4-10*a^6*c^2+c^8-10*b^2*c^6+4*a^8+4*a^2*b^2*c^4-10*b^6*c^2-4*a^2*c^6+9*a^4*c^4+4*a^2*b^4*c^2+b^8-10*a^6*b^2-4*a^2*b^6)*x^2*y^3*z^3-(3*a^6*c^2-9*a^4*c^4+9*a^2*c^6+12*b^2*c^6+7*a^6*b^2-4*b^6*c^2-10*b^4*c^4-21*a^4*b^4+5*b^8-17*a^2*b^2*c^4+9*a^2*b^6-3*c^8-a^2*b^4*c^2-2*a^4*b^2*c^2)*x^2*y^2*z^4-2*(a^2+2*b^2-c^2)*(a^2+b^2-c^2)^2*b^2*x^2*y*z^5-(a^6-3*a^4*b^2+9*a^4*c^2+3*a^2*b^4-18*a^2*b^2*c^2+7*a^2*c^4-b^6+9*b^4*c^2-15*b^2*c^4+7*c^6)*a^2*x*y^5*z^2+(3*a^6-11*a^4*b^2+a^4*c^2+13*a^2*b^4+2*a^2*b^2*c^2+9*a^2*c^4-5*b^6-3*b^4*c^2+21*b^2*c^4-13*c^6)*a^2*x*y^4*z^3+(3*a^6+a^4*b^2-11*a^4*c^2+9*a^2*b^4+2*a^2*b^2*c^2+13*a^2*c^4-13*b^6+21*b^4*c^2-3*b^2*c^4-5*c^6)*a^2*x*y^3*z^4-(a^6+9*a^4*b^2-3*a^4*c^2+7*a^2*b^4-18*a^2*b^2*c^2+3*a^2*c^4+7*b^6-15*b^4*c^2+9*b^2*c^4-c^6)*a^2*x*y^2*z^5-2*(a^2-b^2+c^2)^2*a^4*y^5*z^3+4*(a^2+b^2-c^2)*(a^2-b^2+c^2)*a^4*y^4*z^4-2*(a^2+b^2-c^2)^2*a^4*y^3*z^5)*x : :
Qnh(P ∈ circumcircle) = Midpoint(P, O); Qnh(P ∈ Linf) = P
Some other ETC pairs: (4,5)
------
Qhn(P) = (Haà Na) = x/(-x^2*SA*(2*a^2*y*z+x*(b^2*z+c^2*y))+SB*c^2*x^2*y^2+SC*b^2*x^2*z^2-(S^2-(2*SB+SC)*SB)*x*y^2*z-(S^2-(SB+2*SC)*SC)*x*y*z^2+2*a^4*y^2*z^2) : :
Qhn(P ∈ circumcircle) = P; Qhn(P ∈ Linf) = P
Some other ETC pairs: (1,1320), (2,5503), (3,5504), (6,5505), (67,5505), (79,6595), (80,1320), (265,5504), (671,5503)
Parallelogic centers:
Zon(P) = (Oaà Na) = (a^2-b^2+c^2)*c^6*x^4*y^4+(a^2-b^2+c^2)*(-a^2+b^2+c^2)*c^4*x^4*y^3*z-(3*a^4*b^4+a^8-3*a^6*c^2-b^6*c^2-a^2*c^6-3*a^6*b^2+4*a^4*b^2*c^2-a^2*b^6+2*b^4*c^4+3*a^4*c^4-b^2*c^6)*x^4*y^2*z^2+(a^2+b^2-c^2)*(-a^2+b^2+c^2)*b^4*x^4*y*z^3+(a^2+b^2-c^2)*b^6*x^4*z^4+(a^2-b^2+c^2)^2*c^4*x^3*y^4*z-(-6*a^6*b^2+3*a^4*c^4-c^8+6*a^4*b^4-2*a^2*c^6+3*b^4*c^4+4*a^2*b^4*c^2+2*a^8-2*a^6*c^2-2*b^6*c^2-2*a^2*b^6)*x^3*y^3*z^2-(6*a^4*c^4-2*b^2*c^6-2*a^2*c^6-b^8-2*a^6*b^2-6*a^6*c^2+2*a^8+4*a^2*b^2*c^4-2*a^2*b^6+3*b^4*c^4+3*a^4*b^4)*x^3*y^2*z^3+(a^2+b^2-c^2)^2*b^4*x^3*y*z^4+(-a^2+b^2+c^2)*(a^2+a*c-b^2+c^2)*(a^2-a*c-b^2+c^2)*a^2*x^2*y^4*z^2+(-a^2+b^2+c^2)*(5*a^4-a^2*b^2-a^2*c^2+2*b^4-4*b^2*c^2+2*c^4)*a^2*x^2*y^3*z^3+(-a^2+b^2+c^2)*(a^2+a*b+b^2-c^2)*(a^2-a*b+b^2-c^2)*a^2*x^2*y^2*z^4-(3*a^4-4*a^2*b^2+b^4-2*b^2*c^2+c^4)*a^4*x*y^4*z^3-(3*a^4-4*a^2*c^2+b^4-2*b^2*c^2+c^4)*a^4*x*y^3*z^4-2*a^8*y^4*z^4 : :
Zon(P ∈ circumcircle) = X(3);
Some other ETC pairs: (1,104), (2,14830), (3,13289), (6,32305), (30,32417), (67,3818), (80,4), (265,18381), (671,3830)
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Zno = (Naà Oa) = x^2*(SB*c^2*y^2+z^2*b^2*SC)-2*(S^2+SB*SC)*x^2*y*z-2*x*y*z*SC*SB*(y+z)-a^4*y^2*z^2 : :
Zno(P ∈ circumcircle) = Reflection of O in the midpoint of {P, N} ;
Some other ETC pairs: (1,946), (2,3845), (3,20299), (4,5), (5,20414), (6,19130), (7,20330), (13,13), (14,14), (20,20329), (54,33332), (74,10264), (80,10265), (100,11698), (104,1484), (251,33334), (476,18319), (930,14072), (943,33335)
More:
Qon(X(20)) = X(122)X(154) ∩ X(5878)X(10745)
= SA*(S^2-2*SB*SC)*((8*R^2-SA-3*SW)*S^4+(1536*R^6-16*(11*SA+53*SW)*R^4+8*(9*SA+19*SW)*SW*R^2-(2*SA^2+5*SA*SW+9*SW^2)*SW)*S^2-4*(4*R^2-SW)*(20*R^2-3*SW)*SB*SC*SW) : :
= on lines: {122, 154}, {5878, 10745}
= [ 4.8312316916821620, 3.9298311965133740, -1.3097871272243390 ]
Qho(X(20)) = X(2)X(31361) ∩ X(3)X(16251)
= SC*SB*(S^2-4*SA*SB)*(S^2-4*SA*SC)*(S^2-2*SB*SC) : :
= 3*X(2)-4*X(33531)
= on lines: {2, 31361}, {3, 16251}, {4, 1192}, {30, 3346}, {253, 3146}, {1249, 5895}, {1294, 3529}, {1503, 22049}, {2777, 3183}, {3668, 9579}, {6616, 10152}, {14249, 15005}, {15319, 15682}
= anticomplement of the anticomplement of X(33531)
= polar conjugate of X(14572)
= {-21.4228366136794, -22.8790374979080, 29.3674611867727}
Qoh(X(2)) = X(597)X(11165) ∩ X(1351)X(8724)
= 27*S^4-3*(9*SB*SC+8*SW^2)*S^2-4*(3*SA+SW)*(3*SA-2*SW)*SW^2 : :
= on lines: {597, 11165}, {1351, 8724}, {5304, 9741}, {8359, 32872}, {11148, 27088}, {15271, 32457}, {18841, 33237}
= {-63.8947702703100, 36.7884147942357, 7.66165590296424}
Qnh(X(1)) = X(1)X(474) ∩ X(5)X(519)
= a*(2*a^2-(b+c)*a-3*b^2+8*b*c-3*c^2)*(-a+b+c) : :
= 5*X(1)-X(2136), 7*X(1)-3*X(3158), 3*X(1)-X(3913), 3*X(551)-X(12640), 7*X(2136)-15*X(3158), 3*X(2136)-5*X(3913), X(2136)+5*X(10912), 9*X(3158)-7*X(3913), 3*X(3158)+7*X(10912), X(3189)-5*X(3623), 9*X(3241)-X(12536), X(3625)-3*X(24386), X(3633)+3*X(24392), 3*X(3753)+X(17648), X(3811)-3*X(10247), X(3913)+3*X(10912), 3*X(5603)-X(32049), X(5881)-3*X(11235), 2*X(8666)-3*X(11260), X(8666)-3*X(22837)
= on lines: {1, 474}, {5, 519}, {8, 1392}, {10, 1387}, {21, 643}, {35, 12653}, {145, 3485}, {210, 5330}, {355, 22835}, {377, 3241}, {392, 5506}, {515, 13463}, {517, 5450}, {518, 1351}, {528, 5882}, {529, 4301}, {551, 12640}, {758, 11278}, {952, 12608}, {956, 30323}, {958, 7962}, {960, 2098}, {999, 8668}, {1012, 7982}, {1054, 15854}, {1071, 6264}, {1212, 4919}, {1319, 4188}, {1385, 2802}, {1386, 15955}, {2099, 11520}, {2170, 30618}, {2551, 4345}, {2646, 3885}, {3169, 3723}, {3189, 3623}, {3244, 11263}, {3555, 11009}, {3625, 24386}, {3632, 23708}, {3633, 10827}, {3635, 4743}, {3698, 17535}, {3746, 19525}, {3811, 10247}, {3877, 5302}, {3884, 15254}, {3890, 16859}, {3893, 4511}, {3916, 16558}, {4018, 11280}, {4051, 6603}, {4640, 5697}, {4662, 4853}, {4857, 13272}, {4866, 15829}, {5119, 19535}, {5603, 32049}, {5694, 23960}, {5734, 6957}, {5844, 10916}, {5881, 11235}, {5901, 10915}, {6762, 11224}, {7991, 11194}, {8715, 15178}, {9802, 11015}, {10165, 32157}, {10179, 19860}, {10698, 11256}, {11236, 11522}, {11375, 12648}, {11523, 16189}, {12541, 20057}, {12625, 17532}, {12629, 12635}, {12650, 15726}, {15347, 16863}, {16173, 17619}, {22836, 33179}, {24160, 24864}, {24390, 25416}, {25405, 25440}, {30147, 31792}
= midpoint of X(i) and X(j) for these {i,j}: {1, 10912}, {3244, 21627}, {3680, 3913}, {7982, 12513}, {10698, 11256}, {12629, 12635}, {12653, 22560}
= reflection of X(i) in X(j) for these (i,j): (8715, 15178), (10915, 5901), (11260, 22837), (12607, 13464), (22836, 33179), (32537, 5)
= X(5878)-of-K798i triangle
= X(10912)-of-anti-Aquila triangle
= X(32537)-of-Johnson triangle
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 11376, 5123), (1320, 4861, 3057)
= {5.32447180478965, 10^-30, 1.18321595661992} (Checked twice!!)
Qnh(X(2)) = X(5)X(7615) ∩ X(3589)X(12040)
= 18*S^4-9*(3*SA+4*SW)*(SA-SW)*S^2+4*(3*SA+SW)*(3*SA-2*SW)*SW^2 : :
= on lines: {5, 7615}, {3589, 12040}, {6722, 16509}, {11165, 14535}, {18841, 33237}, {20112, 32538}
= reflection of X(32538) in X(20112)
= {30..084915573808818523, -13.226767339657404820, -1.0876883939337953398}
Qhn(X(20)) = X(4)X(3184) ∩ X(2777)X(3346)
= (S^2-2*SB*SC)*(S^2+(4*R^2-SW)*(48*R^2+SB-8*SW))*(S^2+(4*R^2-SW)*(48*R^2-SA-SB-7*SW)) : :
= on lines: {4, 3184}, {2777, 3346}, {10152, 27089}, {15318, 23241}
= antigonal conjugate of the polar conjugate of X(14572)
= {29.065585003209835506, 28.460122466877527020, -29.477382612027679067}
Zon(X(8)) = X(3)X(119) ∩ X(4)X(1387)
= a^10-2*(b+c)*a^9-(b^2-13*b*c+c^2)*a^8+(b-4*c)*(4*b-c)*(b+c)*a^7-2*(b^4+c^4+3*(2*b^2-7*b*c+2*c^2)*b*c)*a^6+2*(b+c)*(11*b^2-23*b*c+11*c^2)*b*c*a^5+(2*b^4+2*c^4-9*(b^2+4*b*c+c^2)*b*c)*(b-c)^2*a^4-(b^2-c^2)*(b-c)*(4*b^4+4*c^4+(b^2-28*b*c+c^2)*b*c)*a^3+(b^2-c^2)^2*(b^4+c^4+2*(5*b^2-13*b*c+5*c^2)*b*c)*a^2+2*(b^2-c^2)^3*(b-c)*(b^2-4*b*c+c^2)*a-(b^2-c^2)^4*(b-c)^2 : :
= 3*X(5658)-X(6224)
= on lines: {3, 119}, {4, 1387}, {80, 971}, {84, 12619}, {104, 1532}, {144, 153}, {495, 10728}, {515, 10738}, {517, 12641}, {952, 12667}, {1319, 1538}, {1537, 12115}, {2771, 18239}, {2800, 6259}, {3057, 12763}, {5281, 6938}, {5658, 6224}, {5691, 13273}, {5787, 6246}, {6001, 19914}, {6223, 12247}, {6260, 6265}, {6713, 18516}, {6834, 12248}, {6929, 22799}, {10057, 12688}, {10073, 12680}, {10085, 20118}, {11570, 12678}, {12679, 12758}, {12762, 16139}
= reflection of X(i) in X(j) for these (i,j): (84, 12619), (6265, 6260), (10738, 12761), (10742, 6256)
= {X(153), X(6925)}-harmonic conjugate of X(1145)
= {63.148699849659884187, 72.993973790391046124, -76.039178842052445531}
Zno(X(8)) = X(3)X(1610) ∩ X(4)X(653)
= 2*(b+c)*a^6-3*(b^2+c^2)*a^5-(b+c)*(3*b^2-8*b*c+3*c^2)*a^4+2*(3*b^2+5*b*c+3*c^2)*(b-c)^2*a^3-6*(b^2-c^2)*(b-c)*b*c*a^2-(b^2-c^2)^2*(3*b^2-2*b*c+3*c^2)*a+(b^2-c^2)^3*(b-c) : :
= X(3)-3*X(14647), 3*X(5)-2*X(12608), X(1490)-3*X(26446), X(3146)+3*X(14646), 3*X(3679)+X(10864), 3*X(5587)-X(6259), 3*X(5587)+X(7992), 3*X(5657)+X(9799), 3*X(5658)-7*X(9780), 3*X(5770)-X(22770), 3*X(5790)-X(12667), 3*X(5790)+X(12684), 5*X(5818)-X(6223), 3*X(5886)-X(7971), X(8158)-3*X(24477), X(12608)-3*X(12616)
= on lines: {3, 1610}, {4, 653}, {5, 3812}, {8, 6244}, {10, 971}, {12, 15071}, {30, 1158}, {40, 3358}, {46, 20420}, {56, 13226}, {63, 31799}, {65, 8727}, {84, 355}, {140, 6261}, {165, 21677}, {484, 6253}, {495, 1071}, {496, 12672}, {515, 550}, {517, 6245}, {758, 5763}, {942, 20330}, {946, 31794}, {952, 3913}, {1210, 7956}, {1329, 31803}, {1385, 6705}, {1387, 10785}, {1484, 2800}, {1490, 26446}, {1519, 10593}, {1709, 1837}, {1737, 12688}, {1768, 7354}, {1788, 19541}, {1854, 15252}, {2096, 9655}, {2551, 5779}, {2801, 12607}, {3146, 14646}, {3256, 10950}, {3295, 5768}, {3339, 5805}, {3419, 31777}, {3427, 5770}, {3452, 31821}, {3560, 12330}, {3679, 10864}, {3820, 5777}, {4847, 31798}, {5128, 5691}, {5252, 10085}, {5450, 32613}, {5499, 18242}, {5587, 6259}, {5657, 9799}, {5658, 9780}, {5722, 12705}, {5730, 6890}, {5731, 19535}, {5789, 19843}, {5790, 12667}, {5791, 30503}, {5806, 21628}, {5818, 6223}, {5884, 6147}, {5886, 7971}, {5887, 6922}, {5927, 24982}, {6256, 18357}, {6260, 9956}, {6907, 12664}, {6911, 18237}, {7681, 10265}, {7995, 9581}, {8158, 24477}, {8582, 10157}, {9578, 30304}, {9711, 15064}, {9952, 10573}, {10039, 12680}, {10167, 24987}, {10826, 12679}, {10827, 12678}, {10896, 12832}, {11496, 12433}, {11681, 13257}, {12520, 26066}, {12528, 17757}, {12619, 31828}, {13243, 20060}, {15251, 17054}, {18249, 31658}, {19925, 22792}, {24470, 26332}, {25466, 31657}, {31419, 31788}, {31870, 33335}
= midpoint of X(i) and X(j) for these {i,j}: {10, 9948}, {40, 5787}, {84, 355}, {6259, 7992}, {12667, 12684}
= reflection of X(i) in X(j) for these (i,j): (5, 12616), (1385, 6705), (6256, 18357), (6260, 9956), (6261, 140), (22792, 19925)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1210, 9856, 7956), (3812, 12617, 5)
= {-26.677470124308694210, -32.185176066824906201, 38.235387970159480724}
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