HYACINTHOS 26850

[APH]:

Let ABC be a triangle, P a point and PaPbPc the pedal triangle of P.

Denote:

Pab, Pac = the orthogonal projections of Pa on OB,OC, resp.

(N1) = the NPC of PaPabPac. Similarly (N2),(N3)

Ra = the radical axis of (N2),(N3. Similarly Rb, Rc.

Sa = the parallel to Ra through A. Similarly Sb, Sc

1. Which is the locus of P such that Sa,Sb,Sc are concurrent? The Euler Line + + ??

2. Let P be a point on the Euler Line.

2.1. Which is the locus of the radical center P' of (N1),(N2),(N3) [point of concurrence of Ra,Rb,Rc] as P moves on the Euler line?

2.2. Which is the locus of the point of concurrence P" of Sa,Sb,Sc (if concur) as P moves on the Euler line?

[...]
Antreas P. Hatzipolakis


[Peter Moses]:

1: Euler line and this conic:
2 a^2 (a^2 - b^2 - c^2) (a^6 b^2 - 3 a^4 b^4 + 3 a^2 b^6 - b^8 + a^6 c^2 + 2 a^2 b^4 c^2 - 3 b^6 c^2 - 3 a^4 c^4 + 2 a^2 b^2 c^4 + 8 b^4 c^4 + 3 a^2 c^6 - 3 b^2 c^6 - c^8) x^2 + (3 a^12 - 11 a^10 b^2 + 15 a^8 b^4 - 10 a^6 b^6 + 5 a^4 b^8 - 3 a^2 b^10 + b^12 - 11 a^10 c^2 + 21 a^8 b^2 c^2 + 6 a^6 b^4 c^2 - 33 a^4 b^6 c^2 + 19 a^2 b^8 c^2 - 2 b^10 c^2 + 15 a^8 c^4 + 6 a^6 b^2 c^4 + 56 a^4 b^4 c^4 - 16 a^2 b^6 c^4 - b^8 c^4 - 10 a^6 c^6 - 33 a^4 b^2 c^6 - 16 a^2 b^4 c^6 + 4 b^6 c^6 + 5 a^4 c^8 + 19 a^2 b^2 c^8 - b^4 c^8 - 3 a^2 c^10 - 2 b^2 c^10 + c^12) y z + cyclic
 
2.1: A nasty cubic.
 
2.2: circumconic thru X(3519)
(b^2-c^2) (a^2-b^2-c^2) (a^8-3 a^6 b^2+4 a^4 b^4-3 a^2 b^6+b^8-3 a^6 c^2-11 a^4 b^2 c^2+3 a^2 b^4 c^2-4 b^6 c^2+4 a^4 c^4+3 a^2 b^2 c^4+6 b^4 c^4-3 a^2 c^6-4 b^2 c^6+c^8) y z + cyclic

****

Suppose we parameterize a point on the Euler lines as a^2 SA + k SB SC::, then the concurrence is
1 / (a^2 SA (S^2 + 5 SA^2) + k (3 S^2 - SA^2) SB SC)::
 
Thus:
1): L, k = -1
concurrence = 1/(a^2 SA (S^2+5 SA^2)-(3 S^2-SA^2) SB SC)::
 
2): O, k = 0
concurrence = 1/(a^2 SA (S^2+5 SA^2)):: on lines {{4,3521},{93,403},...}
 
3): G, k = 1
concurrence = 1/(a^2 SA (S^2+5 SA^2)+(3 S^2-SA^2) SB SC)::
 
4): N, k = 2
concurrence = 1/(a^2 SA (S^2+5 SA^2)+2 (3 S^2-SA^2) SB SC)::
 
5): H, k = Infinity
concurrence = 1/((3 S^2-SA^2) SB SC):: = X(3519).
 
6): Schiffler, k = R/(r+R)
concurrence = 1/(a^2 (r+R) SA (S^2+5 SA^2)+R (3 S^2-SA^2) SB SC)::
 
Best regards,
Peter.

[Peter Moses]:


Hi Antreas,

Some details for those, unfortunately mostly uninspiring, points ... plus one other:

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L, k=-1; (a^8 - 3*a^6*b^2 + 4*a^4*b^4 - 3*a^2*b^6 + b^8 - 4*a^6*c^2 - 6*a^4*b^2*c^2 - 6*a^2*b^4*c^2 - 4*b^6*c^2 + 6*a^4*c^4 + 13*a^2*b^2*c^4 + 6*b^4*c^4 - 4*a^2*c^6 - 4*b^2*c^6 + c^8)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 3*a^6*c^2 - 6*a^4*b^2*c^2 + 13*a^2*b^4*c^2 - 4*b^6*c^2 + 4*a^4*c^4 - 6*a^2*b^2*c^4 + 6*b^4*c^4 - 3*a^2*c^6 - 4*b^2*c^6 + c^8):: 

on lines {}.
alternative barycentrics: 1/(a^2 SA (S^2+5 SA^2)-(3 S^2-SA^2) SB SC).

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O, k=0; b^2*c^2*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^4 + 3*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 + 3*a^2*c^2 - 2*b^2*c^2 + c^4):: 

on lines {{4, 3521}, {93, 403}, {186, 1105}, {235, 6344}, {1217, 7505}, {1300, 3518}, {8884, 11815}}.
X(i)-isoconjugate of X(j) for these (i,j): {{255, 3520}, {2169, 11591}}.
barycentric product X(i)X(j) for these {i,j}: {{324, 11815}, {2052, 3521}}.
barycentric quotient X(i)/X(j) for these {i,j}: {{53, 11591}, {393, 3520}, {3521, 394}, {11815, 97}}.
major center: Tan[A] / (2 Cos[2 A] + 3).
alternative barycentrics: 1/(a^2 SA (S^2+5 SA^2)).

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G, k=1; (a^8 - 3*a^6*b^2 + 4*a^4*b^4 - 3*a^2*b^6 + b^8 + 10*a^4*b^2*c^2 + 10*a^2*b^4*c^2 - 6*a^4*c^4 - 15*a^2*b^2*c^4 - 6*b^4*c^4 + 8*a^2*c^6 + 8*b^2*c^6 - 3*c^8)*(a^8 - 6*a^4*b^4 + 8*a^2*b^6 - 3*b^8 - 3*a^6*c^2 + 10*a^4*b^2*c^2 - 15*a^2*b^4*c^2 + 8*b^6*c^2 + 4*a^4*c^4 + 10*a^2*b^2*c^4 - 6*b^4*c^4 - 3*a^2*c^6 + c^8):: 

on lines {}.
alternative barycentrics: 1/(a^2 SA (S^2+5 SA^2)+(3 S^2-SA^2) SB SC).

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N, k=2; (a^8 - 3*a^6*b^2 + 4*a^4*b^4 - 3*a^2*b^6 + b^8 - a^6*c^2 + 6*a^4*b^2*c^2 + 6*a^2*b^4*c^2 - b^6*c^2 - 3*a^4*c^4 - 8*a^2*b^2*c^4 - 3*b^4*c^4 + 5*a^2*c^6 + 5*b^2*c^6 - 2*c^8)*(a^8 - a^6*b^2 - 3*a^4*b^4 + 5*a^2*b^6 - 2*b^8 - 3*a^6*c^2 + 6*a^4*b^2*c^2 - 8*a^2*b^4*c^2 + 5*b^6*c^2 + 4*a^4*c^4 + 6*a^2*b^2*c^4 - 3*b^4*c^4 - 3*a^2*c^6 - b^2*c^6 + c^8):: 

on lines {}.
alternative barycentrics: 1/(a^2 SA (S^2+5 SA^2)+2 (3 S^2-SA^2) SB SC).

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X(3519).

H, k = Infinity; (a^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):: 

on lines {{2, 1493}, {3, 539}, {4, 93}, {5, 1173}, {6, 17}, {20, 13452}, {49, 343}, {54, 140}, {64, 1657}, {65, 2962}, {68, 12606}, {69, 12363}, {70, 14791}, {74, 550}, {185, 14861}, {265, 5562}, {290, 7768}, {340, 8795}, {394, 15317}, {399, 14862}, {511, 13433}, {524, 5576}, {542, 13564}, {567, 12899}, {895, 11585}, {1176, 3564}, {1503, 9935}, {1899, 11577}, {2889, 3448}, {2914, 14643}, {2918, 5898}, {2979, 12291}, {3426, 5073}, {3431, 3523}, {3521, 13754}, {3522, 11270}, {3527, 3574}, {3850, 14483}, {4857, 6286}, {5056, 13565}, {5059, 11738}, {5068, 14491}, {5449, 15002}, {5504, 12359}, {6101, 12226}, {6145, 7574}, {6515, 9827}, {7517, 15069}, {10018, 11597}, {10625, 14864}, {10628, 11744}, {11138, 11600}, {11139, 11601}, {11225, 15047}, {11412, 12280}, {12936, 15232}}.
anticomplement X(1493).
complement X(11271).
midpoint of X(i) and X(j) for these {i,j}: {{2888, 12325}, {11412, 12280}}.
reflection of X(i) in X(j) for these {i,j}: {{195, 1209}, {6243, 6152}, {6288, 2888}, {11271, 1493}, {12226, 6101}, {12316, 3574}, {13431, 12242}, {13432, 13431}}.
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11271, 1493), (17, 18, 2963), (93, 562, 14111), (195, 1656, 12242), (195, 13432, 13431), (1209, 12242, 1656), (1209, 13431, 12242), (1656, 13432, 195), (12242, 13431, 195).
on K618.
on Jerabek.
isogonal conjugate of X(3518).
anticomplement of the isogonal of X(1487).
X(i)-anticomplementary conjugate of X(j) for these (i,j): {{1487, 8}, {2962, 2889}}.
X(11140)-Ceva conjugate of X(2963).
X(i)-isoconjugate of X(j) for these (i,j): {{1, 3518}, {4, 2964}, {19, 1994}, {49, 158}, {92, 2965}, {143, 2190}, {162, 1510}, {1973, 7769}, {2148, 14129}, {2167, 14577}, {2216, 6152}}.
cevapoint of X(i) and X(j) for these (i,j): {{633, 634}, {14813, 14814}}.
barycentric product X(i)X(j) for these {i,j}: {{3, 11140}, {63, 2962}, {69, 2963}, {93, 394}, {252, 343}, {525, 930}}.
barycentric quotient X(i)/X(j) for these {i,j}: {{3, 1994}, {5, 14129}, {6, 3518}, {17, 472}, {18, 473}, {48, 2964}, {51, 14577}, {69, 7769}, {93, 2052}, {184, 2965}, {216, 143}, {252, 275}, {562, 14165}, {570, 6152}, {577, 49}, {647, 1510}, {930, 648}, {2962, 92}, {2963, 4}, {8603, 10633}, {8604, 10632}, {11140, 264}, {14111, 11547}}.
major center: 1/(Cot[A] - 3 Tan[A]).
alternative barycentrics: 1/((3 S^2-SA^2) SB SC).

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Schiffler, k = R/(r+R);  b*c*(-(a^9*b) + a^8*b^2 + 4*a^7*b^3 - 4*a^6*b^4 - 6*a^5*b^5 + 6*a^4*b^6 + 4*a^3*b^7 - 4*a^2*b^8 - a*b^9 + b^10 + a^9*c + a^8*b*c - 2*a^7*b^2*c - 2*a^6*b^3*c + 2*a^3*b^6*c + 2*a^2*b^7*c - a*b^8*c - b^9*c - 3*a^7*b*c^2 + 5*a^6*b^2*c^2 + 8*a^5*b^3*c^2 - 14*a^4*b^4*c^2 - 7*a^3*b^5*c^2 + 13*a^2*b^6*c^2 + 2*a*b^7*c^2 - 4*b^8*c^2 - 3*a^7*c^3 + 3*a^6*b*c^3 + 2*a^5*b^2*c^3 - a^3*b^4*c^3 - 7*a^2*b^5*c^3 + 2*a*b^6*c^3 + 4*b^7*c^3 + 8*a^4*b^2*c^4 - 14*a^2*b^4*c^4 + 6*b^6*c^4 + 4*a^5*c^5 + 2*a^3*b^2*c^5 + 8*a^2*b^3*c^5 - 6*b^5*c^5 + 3*a^3*b*c^6 + 5*a^2*b^2*c^6 - 2*a*b^3*c^6 - 4*b^4*c^6 - 3*a^3*c^7 - 3*a^2*b*c^7 - 2*a*b^2*c^7 + 4*b^3*c^7 + a*b*c^8 + b^2*c^8 + a*c^9 - b*c^9)*(-(a^9*b) + 3*a^7*b^3 - 4*a^5*b^5 + 3*a^3*b^7 - a*b^9 + a^9*c - a^8*b*c + 3*a^7*b^2*c - 3*a^6*b^3*c - 3*a^3*b^6*c + 3*a^2*b^7*c - a*b^8*c + b^9*c - a^8*c^2 + 2*a^7*b*c^2 - 5*a^6*b^2*c^2 - 2*a^5*b^3*c^2 - 8*a^4*b^4*c^2 - 2*a^3*b^5*c^2 - 5*a^2*b^6*c^2 + 2*a*b^7*c^2 - b^8*c^2 - 4*a^7*c^3 + 2*a^6*b*c^3 - 8*a^5*b^2*c^3 - 8*a^2*b^5*c^3 + 2*a*b^6*c^3 - 4*b^7*c^3 + 4*a^6*c^4 + 14*a^4*b^2*c^4 + a^3*b^3*c^4 + 14*a^2*b^4*c^4 + 4*b^6*c^4 + 6*a^5*c^5 + 7*a^3*b^2*c^5 + 7*a^2*b^3*c^5 + 6*b^5*c^5 - 6*a^4*c^6 - 2*a^3*b*c^6 - 13*a^2*b^2*c^6 - 2*a*b^3*c^6 - 6*b^4*c^6 - 4*a^3*c^7 - 2*a^2*b*c^7 - 2*a*b^2*c^7 - 4*b^3*c^7 + 4*a^2*c^8 + a*b*c^8 + 4*b^2*c^8 + a*c^9 + b*c^9 - c^10):: 

on lines {}.
alternative barycentrics: 1/(a^2 (r+R) SA (S^2+5 SA^2)+R (3 S^2-SA^2) SB SC).

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X(13597).

X(30), k=-2; (a^8 - 3*a^6*b^2 + 4*a^4*b^4 - 3*a^2*b^6 + b^8 - 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - 3*b^6*c^2 + 3*a^4*c^4 + 6*a^2*b^2*c^4 + 3*b^4*c^4 - a^2*c^6 - b^2*c^6)*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - 3*a^6*c^2 - 2*a^4*b^2*c^2 + 6*a^2*b^4*c^2 - b^6*c^2 + 4*a^4*c^4 - 2*a^2*b^2*c^4 + 3*b^4*c^4 - 3*a^2*c^6 - 3*b^2*c^6 + c^8):: 

on lines {{4,11792},{25,13507},{30, 11703},{99,1232},{110,140},{ 112,6748},{476,5899},{953, 5957},{2687,5959},{2699,5958}} .
reflection of X(4) in X(11792).
on circumcircle.
isogonal conjugate of X(13391).
Collings transform of X(11792).
alternative barycentrics: 1/(a^2 SA (S^2+5 SA^2)-2 (3 S^2-SA^2) SB SC)).

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Best regards,
Peter.