HYACINTHOS 28866

[Kadir Altintas]:

 

Let ABC be a triangle, DEF the pedal triangle of the incenter I = X(1) and P be a point with barycentric coordinates (x:y:z)

Let K1, K2, K3 be the symmedian points of PEF, PFD, PDE respectively

The lines AK1, BK2, CK3 concur at point with first barycentrics

 

Q = 4 (b-c)^2 (b+c) x (x+y+z)-2 a^2 (x+y+z) (c (x+y-z)+b (x-y+z))+a^3 (3 x^2-(y-z)^2+2 x (y+z))-a (-2 b c (3 x^2+4 x (y+z)+(y+z)^2)+b^2 (5 x^2+(y+z)^2+2 x (3 y+z))+c^2 (5 x^2+(y+z)^2+2 x (y+3 z))) : :

 

P=X(1)  ---> Q=X(1)

P=X(2)  ---> Q=X(22166)

P=X(3)  ---> Q=?

P=X(4)  ---> Q=?

P=X(5)  ---> Q=?

P=X(6)  ---> Q=?

P=X(7)  ---> Q=X(20121)

P=X(8)  ---> Q=X(4691)

P=X(11) ---> Q=X(11)

 

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[Ercole Suppa]

 

Let Q = Q(P) the perspector of ABC and K1K2K3

 

*** Q= 4 (b-c)^2 (b+c) x (x+y+z)-2 a^2 (x+y+z) (c (x+y-z)+b (x-y+z))+a^3 (3 x^2-(y-z)^2+2 x (y+z))-a (-2 b c (3 x^2+4 x (y+z)+(y+z)^2)+b^2 (5 x^2+(y+z)^2+2 x (3 y+z))+c^2 (5 x^2+(y+z)^2+2 x (y+3 z))) : :  (barys)

 

*** Some points Q = Q(P):

 

 

--- Q(X(3)) = X(1)X(3) ∩ X(63)X(10090)

 

= a (3 a^6-2 a^5 b-7 a^4 b^2+4 a^3 b^3+5 a^2 b^4-2 a b^5-b^6-2 a^5 c+6 a^4 b c-8 a^2 b^3 c+2 a b^4 c+2 b^5 c-7 a^4 c^2+10 a^2 b^2 c^2+b^4 c^2+4 a^3 c^3-8 a^2 b c^3-4 b^3 c^3+5 a^2 c^4+2 a b c^4+b^2 c^4-2 a c^5+2 b c^5-c^6) : : (barys) 

 

= lies on these lines: {1,3}, {63,10090}, {499,4652}, {5533,9580}, {8070,9579}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,17437,1}, {36,5697,3576}, {36,5902,1420}, {57,14793,1}, {1470,11249,36}, {8071,17700,1}

 

= (6-8-13) search numbers [3.20184701491645961, 2.93842536666211973, 0.128594451949161008]

 

 

--- Q(X(4)) = X(1)X(4) ∩ X(1359)X(13462)

 

= a^7-2 a^6 b-a^5 b^2-a^3 b^4+6 a^2 b^5+a b^6-4 b^7-2 a^6 c+2 a^5 b c+4 a^3 b^3 c-2 a^2 b^4 c-6 a b^5 c+4 b^6 c-a^5 c^2-6 a^3 b^2 c^2-4 a^2 b^3 c^2-a b^4 c^2+12 b^5 c^2+4 a^3 b c^3-4 a^2 b^2 c^3+12 a b^3 c^3-12 b^4 c^3-a^3 c^4-2 a^2 b c^4-a b^2 c^4-12 b^3 c^4+6 a^2 c^5-6 a b c^5+12 b^2 c^5+a c^6+4 b c^6-4 c^7 : : (barys) 

 

= lies on these lnes: {1,4}, {1359,13462}, {3624,24565}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1699,1785,1}

 

= (6-8-13) search numbers [0.768019456221477419, 0.661036277324810023, 2.82855346396498259]

 

 

--- Q(X(5)) = X(1)X(5) ∩ X(3583)X(6950)

 

= 2 a^5 b^2-2 a^4 b^3-4 a^3 b^4+4 a^2 b^5+2 a b^6-2 b^7-2 a^5 b c+6 a^3 b^3 c-2 a^2 b^4 c-4 a b^5 c+2 b^6 c+2 a^5 c^2-5 a^3 b^2 c^2-2 a^2 b^3 c^2-2 a b^4 c^2+6 b^5 c^2-2 a^4 c^3+6 a^3 b c^3-2 a^2 b^2 c^3+8 a b^3 c^3-6 b^4 c^3-4 a^3 c^4-2 a^2 b c^4-2 a b^2 c^4-6 b^3 c^4+4 a^2 c^5-4 a b c^5+6 b^2 c^5+2 a c^6+2 b c^6-2 c^7 : : (barys) 

 

= (3 a R^2-4 b R^2+2 a SB-2 b SB+c SB+2 a SC+b SC-2 c SC-4 a SW+b SW)S^2 +6 R S^3+2 R S SB SC+b SB SC^2-c SB SC^2-b SB SC SW : : (barys)

 

= lies on these lines: {1,5}, {3583,6950}, {10896,14792}, {17566,25639}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {5,1317,7951}

 

= (6-8-13) search numbers [0.662126157385759379, -0.125192011073358414, 3.42173918616481521]

 

 

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

 

= a (3 a^6-2 a^5 b-3 a^4 b^2+4 a^3 b^3-7 a^2 b^4+6 a b^5-b^6-2 a^5 c+6 a^4 b c-8 a^3 b^2 c+8 a^2 b^3 c-6 a b^4 c+2 b^5 c-3 a^4 c^2-8 a^3 b c^2-2 a^2 b^2 c^2-3 b^4 c^2+4 a^3 c^3+8 a^2 b c^3+4 b^3 c^3-7 a^2 c^4-6 a b c^4-3 b^2 c^4+6 a c^5+2 b c^5-c^6) : : (barys) 

 

= (6 a R^2-2 a SW+2 b SW-c SW)S^2 + (2 R SB SW+2 R SC SW+2 R SW^2) S +2 a SB SC SW+b SB SC SW-2 c SB SC SW+3 b SC^2 SW-3 c SC^2 SW-2 b SB SW^2+c SB SW^2-2 b SC SW^2+c SC SW^2-a SW^3 : : (barys)

 

= lies on this line: {1,6}

 

= (6-8-13) search numbers [0.812665225615587883, 1.43742824123153197, 2.27044559538612410]

 

 

--- Q(X(9)) =  (name pending)

 

= a (3 a^6-10 a^5 b+5 a^4 b^2+20 a^3 b^3-35 a^2 b^4+22 a b^5-5 b^6-10 a^5 c+30 a^4 b c-28 a^3 b^2 c+20 a^2 b^3 c-26 a b^4 c+14 b^5 c+5 a^4 c^2-28 a^3 b c^2+14 a^2 b^2 c^2+4 a b^3 c^2-11 b^4 c^2+20 a^3 c^3+20 a^2 b c^3+4 a b^2 c^3+4 b^3 c^3-35 a^2 c^4-26 a b c^4-11 b^2 c^4+22 a c^5+14 b c^5-5 c^6) : : (barys) 

 

= lies on this line: {1,6}

 

= (6-8-13) search numbers [3.44088221209903977, 2.19471110174371303, 0.533149621269944036]

 

 

--- Q(X(10)) = X(1)X(2) ∩ X(381)X(18291)

 

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

 

= lies on these lines: {1,2}, {381,18201}, {3953,15079}, {3976,17606}, {7179,24240}, {10175,24216}, {11231,17715}, {17065,24802}, {17601,18527}, {20118,24806}, {21949,24174}

 

= (6-8-13) search numbers [3.15377908041550342, 1.78787321418759784, 0.947315758047348252]

 

 

Best regards

Ercole Suppa