HYACINTHOS 28952

[Kadir Altintas]:

 

Let ABC be a triangle, P a point and DEF the cevian triangle of P.

 

Denote:

 

La = the refllection of the line FE in BC. 

Define Lb,Lc cyclically.

 

A'B'C' = the triangle bounded by La, Lb, Lc

 

Prove: A'B'C', DEF are perspective.

 

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[Ercole Suppa]:

 

(1) Let Q = Q(P) the perspector of A'B'C'and DEF

 

*** Pairs {P = X(i),Q = X(j)} : {2, 2}, {4, 3575}, {254, 8800}, {1113, 1113}, {1114, 1114}

 

*** Some points:

 

 

Q(X(1)) = X(1)X(21) ∩ X(12)X(42)

 

= a (b+c) (a^5-2 a^3 b^2+a b^4-a^3 b c-2 a^2 b^2 c+b^4 c-2 a^3 c^2-2 a^2 b c^2-2 a b^2 c^2-b^3 c^2-b^2 c^3+a c^4+b c^4) : : (barys) 

 

= lies on these  lines: {1,21}, {12,42}, {65,3724}, {78,21020}, {581,24725}, {756,3191}, {899,6668}, {2177,11491}, {2294,9310}, {2646,20718}, {2658,2667}, {3120,11553}, {3720,4999}, {3725,3924}, {4343,5857}, {4647,22836}, {5855,10459}, {17018,20060}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,5496,12081}

 

= (6-9-13) search numbers [1.29514296668583387, 1.52846470234268068, 1.98473831873829223]

 

 

Q(X(3)) = X(3)X(1093) ∩ X(577)X(1204)

 

= a^2 (a^2-b^2-c^2)^2 (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+4 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) (a^16-4 a^14 b^2+6 a^12 b^4-5 a^10 b^6+5 a^8 b^8-6 a^6 b^10+4 a^4 b^12-a^2 b^14-4 a^14 c^2+15 a^12 b^2 c^2-17 a^10 b^4 c^2+a^8 b^6 c^2+10 a^6 b^8 c^2-7 a^4 b^10 c^2+3 a^2 b^12 c^2-b^14 c^2+6 a^12 c^4-17 a^10 b^2 c^4+20 a^8 b^4 c^4-4 a^6 b^6 c^4-8 a^4 b^8 c^4-3 a^2 b^10 c^4+6 b^12 c^4-5 a^10 c^6+a^8 b^2 c^6-4 a^6 b^4 c^6+22 a^4 b^6 c^6+a^2 b^8 c^6-15 b^10 c^6+5 a^8 c^8+10 a^6 b^2 c^8-8 a^4 b^4 c^8+a^2 b^6 c^8+20 b^8 c^8-6 a^6 c^10-7 a^4 b^2 c^10-3 a^2 b^4 c^10-15 b^6 c^10+4 a^4 c^12+3 a^2 b^2 c^12+6 b^4 c^12-a^2 c^14-b^2 c^14) : : (barys) 

 

= S^6 + (-SB SC+8 R^2 SW-2 SW^2)S^4 + (-1536 R^8+64 R^6 SB+64 R^6 SC+16 R^4 SB SC+1344 R^6 SW-32 R^4 SB SW-32 R^4 SC SW-12 R^2 SB SC SW-432 R^4 SW^2+4 R^2 SB SW^2+4 R^2 SC SW^2+2 SB SC SW^2+60 R^2 SW^3-3 SW^4)S^2 +2048 R^8 SB SC-1664 R^6 SB SC SW+496 R^4 SB SC SW^2-64 R^2 SB SC SW^3+3 SB SC SW^4 : : (barys)  

 

= lies on these lines: {3,1093}, {577,1204}

 

= (6-9-13) search numbers [-72.9703061259603234, -68.1885250809761897, 84.5267077499495828]

 

 

Q(X(5)) = (name pending)

 

= (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4)^2 (2 a^6-3 a^4 b^2+b^6-3 a^4 c^2-b^4 c^2-b^2 c^4+c^6) (a^20-4 a^18 b^2+a^16 b^4+20 a^14 b^6-42 a^12 b^8+28 a^10 b^10+14 a^8 b^12-36 a^6 b^14+25 a^4 b^16-8 a^2 b^18+b^20-4 a^18 c^2+8 a^16 b^2 c^2+10 a^14 b^4 c^2-25 a^12 b^6 c^2-19 a^10 b^8 c^2+64 a^8 b^10 c^2-28 a^6 b^12 c^2-25 a^4 b^14 c^2+25 a^2 b^16 c^2-6 b^18 c^2+a^16 c^4+10 a^14 b^2 c^4-5 a^12 b^4 c^4-51 a^10 b^6 c^4+41 a^8 b^8 c^4+52 a^6 b^10 c^4-55 a^4 b^12 c^4-3 a^2 b^14 c^4+10 b^16 c^4+20 a^14 c^6-25 a^12 b^2 c^6-51 a^10 b^4 c^6+50 a^8 b^6 c^6+12 a^6 b^8 c^6+45 a^4 b^10 c^6-61 a^2 b^12 c^6+10 b^14 c^6-42 a^12 c^8-19 a^10 b^2 c^8+41 a^8 b^4 c^8+12 a^6 b^6 c^8+20 a^4 b^8 c^8+47 a^2 b^10 c^8-59 b^12 c^8+28 a^10 c^10+64 a^8 b^2 c^10+52 a^6 b^4 c^10+45 a^4 b^6 c^10+47 a^2 b^8 c^10+88 b^10 c^10+14 a^8 c^12-28 a^6 b^2 c^12-55 a^4 b^4 c^12-61 a^2 b^6 c^12-59 b^8 c^12-36 a^6 c^14-25 a^4 b^2 c^14-3 a^2 b^4 c^14+10 b^6 c^14+25 a^4 c^16+25 a^2 b^2 c^16+10 b^4 c^16-8 a^2 c^18-6 b^2 c^18+c^20) : : (barys) 

 

= lies on this line: {5,8884}

 

= (6-9-13) search numbers [0.455817736651812713, -0.108330625117599397, 3.50528519007264974]

 

 

Q(X(6)) = (name pending)

 

= a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^8-4 a^6 b^2+4 a^4 b^4-a^2 b^6-4 a^6 c^2+7 a^4 b^2 c^2+a^2 b^4 c^2-b^6 c^2+4 a^4 c^4+a^2 b^2 c^4+2 b^4 c^4-a^2 c^6-b^2 c^6) : : (barys) 

 

= (8 R^2-5 SB-5 SC)S^4 + (-64 R^4 SB-64 R^4 SC-4 R^2 SB SC-16 R^4 SW+32 R^2 SB SW+32 R^2 SC SW+4 R^2 SW^2-6 SB SW^2-6 SC SW^2)S^2 + 4 R^2 SB SW^3+4 R^2 SC SW^3-SB SW^4-SC SW^4 : : (barys)  

 

= lies on this line: {6,95}

 

= (6-9-13) search numbers [-0.672126749406475774, 2.73410794986569449, 2.05803286249573327]

 

 

Best regards

Ercole Suppa