HYACINTHOS 29372

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:

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

A',B',C' = the midpoints of AP, BP, CP, resp.

A"B"C" = the orthic triangle of NaNbNc.

A'B'C', A"B"C" are perspective.

Perspectors for O, I, N ...... ?


[Peter Moses]:

Hi Antreas,

O --> 

 

X(3)X(14111)∩X(5)X(51) =

 

= (a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^10*b^2 - 5*a^8*b^4 + 10*a^6*b^6 - 10*a^4*b^8 + 5*a^2*b^10 - b^12 + a^10*c^2 - 2*a^8*b^2*c^2 + 6*a^4*b^6*c^2 - 9*a^2*b^8*c^2 + 4*b^10*c^2 - 5*a^8*c^4 + 2*a^4*b^4*c^4 + 4*a^2*b^6*c^4 - 7*b^8*c^4 + 10*a^6*c^6 + 6*a^4*b^2*c^6 + 4*a^2*b^4*c^6 + 8*b^6*c^6 - 10*a^4*c^8 - 9*a^2*b^2*c^8 - 7*b^4*c^8 + 5*a^2*c^10 + 4*b^2*c^10 - c^12) : : 

= lies on these lines: {3,14111}, {5,51}

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I --> X(3649).

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N --> 

 

X(5)X(22101)∩X(137)X(32638) = 

 

= 2*a^22 - 21*a^20*b^2 + 98*a^18*b^4 - 267*a^16*b^6 + 468*a^14*b^8 - 546*a^12*b^10 + 420*a^10*b^12 - 198*a^8*b^14 + 42*a^6*b^16 + 7*a^4*b^18 - 6*a^2*b^20 + b^22 - 21*a^20*c^2 + 150*a^18*b^2*c^2 - 430*a^16*b^4*c^2 + 583*a^14*b^6*c^2 - 227*a^12*b^8*c^2 - 405*a^10*b^10*c^2 + 635*a^8*b^12*c^2 - 355*a^6*b^14*c^2 + 52*a^4*b^16*c^2 + 27*a^2*b^18*c^2 - 9*b^20*c^2 + 98*a^18*c^4 - 430*a^16*b^2*c^4 + 624*a^14*b^4*c^4 - 218*a^12*b^6*c^4 - 94*a^10*b^8*c^4 - 353*a^8*b^10*c^4 + 694*a^6*b^12*c^4 - 346*a^4*b^14*c^4 - 10*a^2*b^16*c^4 + 35*b^18*c^4 - 267*a^16*c^6 + 583*a^14*b^2*c^6 - 218*a^12*b^4*c^6 - 100*a^10*b^6*c^6 - 39*a^8*b^8*c^6 - 360*a^6*b^10*c^6 + 656*a^4*b^12*c^6 - 180*a^2*b^14*c^6 - 75*b^16*c^6 + 468*a^14*c^8 - 227*a^12*b^2*c^8 - 94*a^10*b^4*c^8 - 39*a^8*b^6*c^8 - 42*a^6*b^8*c^8 - 369*a^4*b^10*c^8 + 528*a^2*b^12*c^8 + 90*b^14*c^8 - 546*a^12*c^10 - 405*a^10*b^2*c^10 - 353*a^8*b^4*c^10 - 360*a^6*b^6*c^10 - 369*a^4*b^8*c^10 - 718*a^2*b^10*c^10 - 42*b^12*c^10 + 420*a^10*c^12 + 635*a^8*b^2*c^12 + 694*a^6*b^4*c^12 + 656*a^4*b^6*c^12 + 528*a^2*b^8*c^12 - 42*b^10*c^12 - 198*a^8*c^14 - 355*a^6*b^2*c^14 - 346*a^4*b^4*c^14 - 180*a^2*b^6*c^14 + 90*b^8*c^14 + 42*a^6*c^16 + 52*a^4*b^2*c^16 - 10*a^2*b^4*c^16 - 75*b^6*c^16 + 7*a^4*c^18 + 27*a^2*b^2*c^18 + 35*b^4*c^18 - 6*a^2*c^20 - 9*b^2*c^20 + c^22 : : 

= lies on these lines: {5,22101}, {137,32638}, {547,1209}, {18583,20413}

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G --> 

 

X(2)X(22100)∩X(524)X(547) = 

 

= 16*a^10 - 62*a^8*b^2 + 100*a^6*b^4 - 95*a^4*b^6 + 46*a^2*b^8 - 5*b^10 - 62*a^8*c^2 + 22*a^6*b^2*c^2 + 135*a^4*b^4*c^2 - 197*a^2*b^6*c^2 + 22*b^8*c^2 + 100*a^6*c^4 + 135*a^4*b^2*c^4 + 270*a^2*b^4*c^4 - 17*b^6*c^4 - 95*a^4*c^6 - 197*a^2*b^2*c^6 - 17*b^4*c^6 + 46*a^2*c^8 + 22*b^2*c^8 - 5*c^10 : : 

 

= lies on these lines: {2,22100}, {524,547}

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In general P{p,q,r} --> c^2*(a^8 - 6*a^6*b^2 + 12*a^4*b^4 - 10*a^2*b^6 + 3*b^8 - 4*a^6*c^2 + 8*a^4*b^2*c^2 - 4*b^6*c^2 + 6*a^4*c^4 - 2*a^2*b^2*c^4 - 4*a^2*c^6 + c^8)*p^5*q^2 + c^2*(a^8 - 10*a^6*b^2 + 24*a^4*b^4 - 22*a^2*b^6 + 7*b^8 - 4*a^6*c^2 + 4*a^4*b^2*c^2 + 16*a^2*b^4*c^2 - 16*b^6*c^2 + 4*a^4*c^4 - 2*a^2*b^2*c^4 + 10*b^4*c^4 - c^8)*p^4*q^3 - c^2*(a^2 + 5*b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*b^2*c^2 + c^4)*p^3*q^4 - c^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*b^2*c^2 + c^4)*p^2*q^5 + (a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*a^2*b^8 + b^10 + a^8*c^2 - 8*a^6*b^2*c^2 + 16*a^4*b^4*c^2 - 12*a^2*b^6*c^2 + 3*b^8*c^2 - 4*a^6*c^4 + 16*a^4*b^2*c^4 - 4*b^6*c^4 + 6*a^4*c^6 - 12*a^2*b^2*c^6 - 4*b^4*c^6 - 4*a^2*c^8 + 3*b^2*c^8 + c^10)*p^5*q*r + (2*a^10 - 8*a^8*b^2 + 12*a^6*b^4 - 8*a^4*b^6 + 2*a^2*b^8 - 5*a^8*c^2 - 4*a^6*b^2*c^2 + 34*a^4*b^4*c^2 - 36*a^2*b^6*c^2 + 11*b^8*c^2 + 4*a^6*c^4 + 24*a^4*b^2*c^4 + 24*a^2*b^4*c^4 - 20*b^6*c^4 - 2*a^4*c^6 - 16*a^2*b^2*c^6 + 6*b^4*c^6 + 2*a^2*c^8 + 4*b^2*c^8 - c^10)*p^4*q^2*r + 2*(2*a^10 - 10*a^8*b^2 + 20*a^6*b^4 - 20*a^4*b^6 + 10*a^2*b^8 - 2*b^10 - 6*a^8*c^2 + a^6*b^2*c^2 + 27*a^4*b^4*c^2 - 33*a^2*b^6*c^2 + 11*b^8*c^2 + 7*a^6*c^4 + 6*a^4*b^2*c^4 + 31*a^2*b^4*c^4 - 20*b^6*c^4 - 5*a^4*c^6 - 11*a^2*b^2*c^6 + 14*b^4*c^6 + 3*a^2*c^8 - 2*b^2*c^8 - c^10)*p^3*q^3*r + (2*a^10 - 12*a^8*b^2 + 28*a^6*b^4 - 32*a^4*b^6 + 18*a^2*b^8 - 4*b^10 - 9*a^8*c^2 + 46*a^4*b^4*c^2 - 56*a^2*b^6*c^2 + 19*b^8*c^2 + 10*a^6*c^4 - 4*a^4*b^2*c^4 + 62*a^2*b^4*c^4 - 36*b^6*c^4 - 10*a^4*c^6 - 28*a^2*b^2*c^6 + 34*b^4*c^6 + 4*a^2*c^8 - 16*b^2*c^8 + 3*c^10)*p^2*q^4*r - (a^4 - 2*a^2*b^2 + b^4 - 2*b^2*c^2 + c^4)*(a^4*b^2 - 2*a^2*b^4 + b^6 + 3*a^4*c^2 + 4*a^2*b^2*c^2 - 3*b^4*c^2 - 2*a^2*c^4 + 3*b^2*c^4 - c^6)*p*q^5*r + b^2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 6*a^6*c^2 + 8*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 12*a^4*c^4 - 10*a^2*c^6 - 4*b^2*c^6 + 3*c^8)*p^5*r^2 + (2*a^10 - 5*a^8*b^2 + 4*a^6*b^4 - 2*a^4*b^6 + 2*a^2*b^8 - b^10 - 8*a^8*c^2 - 4*a^6*b^2*c^2 + 24*a^4*b^4*c^2 - 16*a^2*b^6*c^2 + 4*b^8*c^2 + 12*a^6*c^4 + 34*a^4*b^2*c^4 + 24*a^2*b^4*c^4 + 6*b^6*c^4 - 8*a^4*c^6 - 36*a^2*b^2*c^6 - 20*b^4*c^6 + 2*a^2*c^8 + 11*b^2*c^8)*p^4*q*r^2 + (8*a^10 - 29*a^8*b^2 + 46*a^6*b^4 - 44*a^4*b^6 + 26*a^2*b^8 - 7*b^10 - 29*a^8*c^2 + 24*a^6*b^2*c^2 + 52*a^4*b^4*c^2 - 68*a^2*b^6*c^2 + 21*b^8*c^2 + 46*a^6*c^4 + 52*a^4*b^2*c^4 + 84*a^2*b^4*c^4 - 14*b^6*c^4 - 44*a^4*c^6 - 68*a^2*b^2*c^6 - 14*b^4*c^6 + 26*a^2*c^8 + 21*b^2*c^8 - 7*c^10)*p^3*q^2*r^2 + (10*a^10 - 41*a^8*b^2 + 70*a^6*b^4 - 64*a^4*b^6 + 32*a^2*b^8 - 7*b^10 - 33*a^8*c^2 + 30*a^6*b^2*c^2 + 74*a^4*b^4*c^2 - 98*a^2*b^6*c^2 + 27*b^8*c^2 + 52*a^6*c^4 + 36*a^4*b^2*c^4 + 118*a^2*b^4*c^4 - 38*b^6*c^4 - 46*a^4*c^6 - 70*a^2*b^2*c^6 + 22*b^4*c^6 + 18*a^2*c^8 - 3*b^2*c^8 - c^10)*p^2*q^3*r^2 + 2*(2*a^10 - 10*a^8*b^2 + 19*a^6*b^4 - 17*a^4*b^6 + 7*a^2*b^8 - b^10 - 7*a^8*c^2 + 4*a^6*b^2*c^2 + 24*a^4*b^4*c^2 - 26*a^2*b^6*c^2 + 5*b^8*c^2 + 9*a^6*c^4 + 3*a^4*b^2*c^4 + 36*a^2*b^4*c^4 - 10*b^6*c^4 - 10*a^4*c^6 - 22*a^2*b^2*c^6 + 10*b^4*c^6 + 5*a^2*c^8 - 5*b^2*c^8 + c^10)*p*q^4*r^2 - 2*a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^4 - 2*a^2*b^2 + b^4 - 2*b^2*c^2 + c^4)*q^5*r^2 + b^2*(a^8 - 4*a^6*b^2 + 4*a^4*b^4 - b^8 - 10*a^6*c^2 + 4*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 24*a^4*c^4 + 16*a^2*b^2*c^4 + 10*b^4*c^4 - 22*a^2*c^6 - 16*b^2*c^6 + 7*c^8)*p^4*r^3 + 2*(2*a^10 - 6*a^8*b^2 + 7*a^6*b^4 - 5*a^4*b^6 + 3*a^2*b^8 - b^10 - 10*a^8*c^2 + a^6*b^2*c^2 + 6*a^4*b^4*c^2 - 11*a^2*b^6*c^2 - 2*b^8*c^2 + 20*a^6*c^4 + 27*a^4*b^2*c^4 + 31*a^2*b^4*c^4 + 14*b^6*c^4 - 20*a^4*c^6 - 33*a^2*b^2*c^6 - 20*b^4*c^6 + 10*a^2*c^8 + 11*b^2*c^8 - 2*c^10)*p^3*q*r^3 + (10*a^10 - 33*a^8*b^2 + 52*a^6*b^4 - 46*a^4*b^6 + 18*a^2*b^8 - b^10 - 41*a^8*c^2 + 30*a^6*b^2*c^2 + 36*a^4*b^4*c^2 - 70*a^2*b^6*c^2 - 3*b^8*c^2 + 70*a^6*c^4 + 74*a^4*b^2*c^4 + 118*a^2*b^4*c^4 + 22*b^6*c^4 - 64*a^4*c^6 - 98*a^2*b^2*c^6 - 38*b^4*c^6 + 32*a^2*c^8 + 27*b^2*c^8 - 7*c^10)*p^2*q^2*r^3 + 2*a^2*(4*a^8 - 13*a^6*b^2 + 23*a^4*b^4 - 23*a^2*b^6 + 9*b^8 - 13*a^6*c^2 + 10*a^4*b^2*c^2 + 23*a^2*b^4*c^2 - 36*b^6*c^2 + 23*a^4*c^4 + 23*a^2*b^2*c^4 + 54*b^4*c^4 - 23*a^2*c^6 - 36*b^2*c^6 + 9*c^8)*p*q^3*r^3 + 2*a^2*(a^8 - 3*a^6*b^2 + 6*a^4*b^4 - 7*a^2*b^6 + 3*b^8 - 3*a^6*c^2 + 2*a^4*b^2*c^2 + 9*a^2*b^4*c^2 - 12*b^6*c^2 + 4*a^4*c^4 + 3*a^2*b^2*c^4 + 18*b^4*c^4 - 5*a^2*c^6 - 12*b^2*c^6 + 3*c^8)*q^4*r^3 - b^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + 5*c^2)*(a^4 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*p^3*r^4 + (2*a^10 - 9*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 4*a^2*b^8 + 3*b^10 - 12*a^8*c^2 - 4*a^4*b^4*c^2 - 28*a^2*b^6*c^2 - 16*b^8*c^2 + 28*a^6*c^4 + 46*a^4*b^2*c^4 + 62*a^2*b^4*c^4 + 34*b^6*c^4 - 32*a^4*c^6 - 56*a^2*b^2*c^6 - 36*b^4*c^6 + 18*a^2*c^8 + 19*b^2*c^8 - 4*c^10)*p^2*q*r^4 + 2*(2*a^10 - 7*a^8*b^2 + 9*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 + b^10 - 10*a^8*c^2 + 4*a^6*b^2*c^2 + 3*a^4*b^4*c^2 - 22*a^2*b^6*c^2 - 5*b^8*c^2 + 19*a^6*c^4 + 24*a^4*b^2*c^4 + 36*a^2*b^4*c^4 + 10*b^6*c^4 - 17*a^4*c^6 - 26*a^2*b^2*c^6 - 10*b^4*c^6 + 7*a^2*c^8 + 5*b^2*c^8 - c^10)*p*q^2*r^4 + 2*a^2*(a^8 - 3*a^6*b^2 + 4*a^4*b^4 - 5*a^2*b^6 + 3*b^8 - 3*a^6*c^2 + 2*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 12*b^6*c^2 + 6*a^4*c^4 + 9*a^2*b^2*c^4 + 18*b^4*c^4 - 7*a^2*c^6 - 12*b^2*c^6 + 3*c^8)*q^3*r^4 - b^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*p^2*r^5 - (a^4 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(3*a^4*b^2 - 2*a^2*b^4 - b^6 + a^4*c^2 + 4*a^2*b^2*c^2 + 3*b^4*c^2 - 2*a^2*c^4 - 3*b^2*c^4 + c^6)*p*q*r^5 - 2*a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^4 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*q^2*r^5 : :

 

Best regards,
Peter Moses.