HYACINTHOS 29077

[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 of PBC, PCA, PAB, resp.

D = the Poncelet point of ABCP

DNa, DNb, DNc intersect the pedal circle of P (circumcircle of A'B'C') again at A*, B*, C*, resp.

For P = N:

1. ABC, A*B*C* are perspective

2. A'B'C', A*B*C* are perspective.

Perspectors ?


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[Ercole Suppa]:

If P = N we have:

 

1. perspector of ABC and A*B*C* 

Q1 = X(3)X(252) ∩ X(137)X(1487)

= (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6)^2 : : (barys)

= (17 R^4-12 SB SC-4 R^2 SW-4 SW^2)S^4 + (-132 R^8-125 R^6 SB-125 R^6 SC-67 R^4 SB SC+293 R^6 SW+150 R^4 SB SW+150 R^4 SC SW+44 R^2 SB SC SW-225 R^4 SW^2-60 R^2 SB SW^2-60 R^2 SC SW^2-4 SB SC SW^2+72 R^2 SW^3+8 SB SW^3+8 SC SW^3-8 SW^4)S^2 + 12 S^6-18 R^8 SB SC+27 R^6 SB SC SW-9 R^4 SB SC SW^2 : : (barys)

= lies on these lines: {3,252}, {137,1487}, {1157,6592},{3519,14072}, {19268,19552}, {27868,31392}

 

= isogonal conjugate of X(27357)

= barycentric quotient of X(i) and X(j) for these {i,j}: {6,27357}

= trilinear quotient of X(i) and X(j) for these {i,j}: {1,27357}

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {252,930,24144}

= (6-9-13)  search numbers:  [6.32054932120575309492, 5.62123862549858433087, -3.16813886860960378514]


2. perspector of A'B'C' and A*B*C*

Q2 = X(5)X(49) ∩ X(1263)X(15038)

= (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (2 a^16-11 a^14 b^2+25 a^12 b^4-31 a^10 b^6+25 a^8 b^8-17 a^6 b^10+11 a^4 b^12-5 a^2 b^14+b^16-11 a^14 c^2+34 a^12 b^2 c^2-31 a^10 b^4 c^2-4 a^8 b^6 c^2+31 a^6 b^8 c^2-38 a^4 b^10 c^2+27 a^2 b^12 c^2-8 b^14 c^2+25 a^12 c^4-31 a^10 b^2 c^4-5 a^6 b^6 c^4+34 a^4 b^8 c^4-51 a^2 b^10 c^4+28 b^12 c^4-31 a^10 c^6-4 a^8 b^2 c^6-5 a^6 b^4 c^6-14 a^4 b^6 c^6+29 a^2 b^8 c^6-56 b^10 c^6+25 a^8 c^8+31 a^6 b^2 c^8+34 a^4 b^4 c^8+29 a^2 b^6 c^8+70 b^8 c^8-17 a^6 c^10-38 a^4 b^2 c^10-51 a^2 b^4 c^10-56 b^6 c^10+11 a^4 c^12+27 a^2 b^2 c^12+28 b^4 c^12-5 a^2 c^14-8 b^2 c^14+c^16) : : (barys)

= (4 R^2+8 SB+8 SC)S^4 + (-93 R^6+14 R^4 SB+14 R^4 SC+52 R^2 SB SC+76 R^4 SW-8 R^2 SB SW-8 R^2 SC SW-16 SB SC SW-16 R^2 SW^2) S^2 -9 R^6 SB SC : : (barys)

= lies on these lines: {5,49}, {1263,15038}, {6343,13621}, {6592,11584}, {10096,14071}, {14627,31674}, {14857,24306}

= barycentric quotient of X(i) and X(j) for these {i,j}: {24385,13582}

= trilinear product of X(i) and X(j) for these {i,j}: {1749,24385},{1749,24385}

= (6-9-13)  search numbers:  [0.24409911532262817460, 0.18909778573274509247, 3.39708950009741949698]


Best regards,
Ercole Suppa