HYACINTHOS 23101

Antreas P. Hatzipolakis

 

[APH]:

Let P be a point on the Neuberg cubic (K001) and M the intersection
of the Eluler lines L1, L2, L3 of PBC, PCA, PAB, resp
Denote:
C2 = the other than M intersection of the line L2 and the circle (C,CM)
B3 = the other than M intersection of the line L3 and the circle (B,BM)
O1 = the circumcenter of MC2B3
Similarly O2, O3
The triangles ABC, O1O2O3 are cyclologic.

 

 

[Angel Montesdeoca]


*** P=I=X(1), Q=X(3065) are the only two points on the Neuberg cubic
(K001) such that the Euler lines of the six triangles PBC, PCA, PAB,
QBC, QCA, QAB concur at X(21), Schiffler point (on the Euler line of
ABC).

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*** Denote:
C2 = the other than X(21) intersection of the Euler line of ICA and the
circle (C,CX(21))

B3 = the other than X(21) intersection of the Euler line of IAB and the
circle (B,BX(21))

O1 = the circumcenter of X(21)C2B3

O1 =(1:b(a-c)/(a^2-b^2+c^2-a(b+2c):c(a-b)/(a^2+b^2-c^2-a(2b+c)))

Similarly O2, O3
The triangles ABC, O1O2O3 are cyclologic.

The cyclologic center (ABC-O1O2O3) is:
Z =
(a (a^9 + a^7 (-6 b^2 + 7 b c - 6 c^2) - (b - c)^4 (b + c)^3 (2 b^2 +
5 b c + 2 c^2) + a^6 (2 b^3 - 3 b^2 c - 3 b c^2 + 2 c^3) +
a (b^2 - c^2)^2 (3 b^4 - b^3 c + 2 b^2 c^2 - b c^3 + 3 c^4) +
a^5 (12 b^4 - 15 b^3 c + 13 b^2 c^2 - 15 b c^3 + 12 c^4) +
a^4 (-6 b^5 + 3 b^4 c + 5 b^3 c^2 + 5 b^2 c^3 + 3 b c^4 - 6 c^5) +
a^3 (-10 b^6 + 9 b^5 c - 3 b^4 c^2 + 5 b^3 c^3 - 3 b^2 c^4 +
9 b c^5 - 10 c^6) +
a^2 (6 b^7 + 3 b^6 c - 11 b^5 c^2 + b^4 c^3 + b^3 c^4 -
11 b^2 c^5 + 3 b c^6 + 6 c^7)):...:...)

Exact trilinear coordinates of Z at the triangle ETC (6-9-13):
(3.10306953362835054,4.70121811205322202,-1.04621091888093639)

The cyclologic center (O1O2O3-ABC) is X(3065).

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*** Denote:
C2' = the other than X(21) intersection of the Euler line of the
triangle

X(3065)CA and the circle (C,CX(21)).

B3' = the other than X(21) intersection of the Euler line of the
triangle

X(3065)AB and the circle (B,BX(21)).

O1' = the circumcenter of the triangle X(21)C2'B3'.
O1' = (-1:b(a-c)/(a^2-b^2+c^2+a(b-2c)): c(a-b)/(a^2+b^2-c^2+a(c-2b)))

Similarly O2', O3'.
The triangles ABC, O1'O2'O3' are cyclologic.

The cyclologic center (ABC-O1'O2'O3') is X(1320).
The cyclologic center (O1'O2'O3'-ABC) is X(1).

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*** The triangles O1O2O3, O1'O2'O3' are cyclologic.

The cyclologic center (O1O2O3 - O'1O'2O'3):

T = (a (a^9 + 10 a^7 b c - 3 a^8 (b + c) + (b - c)^6 (b + c)^3 -
a (b - c)^4 (b + c)^2 (3 b^2 - 2 b c + 3 c^2) +
a^6 (8 b^3 - 6 b^2 c - 6 b c^2 + 8 c^3) -
a^5 (6 b^4 + 12 b^3 c - 13 b^2 c^2 + 12 b c^3 + 6 c^4) -
a^4 (6 b^5 - 18 b^4 c + 7 b^3 c^2 + 7 b^2 c^3 - 18 b c^4 + 6 c^5) -
a^2 b c (6 b^5 - 13 b^4 c + 11 b^3 c^2 + 11 b^2 c^3 - 13 b c^4 +
6 c^5) +
a^3 (8 b^6 - 6 b^5 c - 9 b^4 c^2 + 26 b^3 c^3 - 9 b^2 c^4 -
6 b c^5 + 8 c^6)) : .... : ...)

Exact trilinear coordinates of T at the triangle ETC (6-9-13):

(-3.57115072621920348, 7.32411086441065946, 0.218349603262785701)

The cyclologic center (O1'O2'O3' - O1O2O3):

T' = (a (a^9 - a^8 (b + c) - (b - c)^4 (b + c)^5 +
a^7 (-4 b^2 + 2 b c - 4 c^2) + a (b - c)^2 (b + c)^4 (b^2 + c^2) +
4 a^6 (b^3 + c^3) +
a^5 (6 b^4 - 2 b^3 c + 9 b^2 c^2 - 2 b c^3 + 6 c^4) +
a^4 (-6 b^5 + 2 b^4 c + b^3 c^2 + b^2 c^3 + 2 b c^4 - 6 c^5) -
a^3 (4 b^6 + 2 b^5 c + 5 b^4 c^2 - 2 b^3 c^3 + 5 b^2 c^4 +
2 b c^5 + 4 c^6) +
a^2 (4 b^7 - 5 b^5 c^2 - 3 b^4 c^3 - 3 b^3 c^4 - 5 b^2 c^5 +
4 c^7)) : ... : ...)

Exact trilinear coordinates of T' at the triangle ETC (6-9-13):

(5.40190694693080237, 1.42496790689825296, 0.160960417009833657)

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*** The triangles O1O2O3, O'1O'2O'3 are "ORTHOLOGIC".

The ORTHOLOGIC center (O1O2O3 - O1'O2'O3'):

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

Exact trilinear coordinates of W at the triangle ETC (6-9-13):

(8.85742096870961083, -0.578609773732795905, -0.0467999679896595013)

The ORTHOLOGIC center (O1'O2'O3' - O1O2O3):

W' = (a^13 - 2 a^12 (b + c) - (b - c)^8 (b + c)^5 +
a^11 (-3 b^2 + 5 b c - 3 c^2) +
2 a (b - c)^6 (b + c)^4 (b^2 - b c + c^2) -
a^8 (b + c) (3 b^2 - 2 b c + 3 c^2)^2 +
a^10 (7 b^3 + 2 b^2 c + 2 b c^2 + 7 c^3) +
a^2 (b - c)^4 (b + c)^3 (3 b^4 - 3 b^3 c - 5 b^2 c^2 - 3 b c^3 +
3 c^4) + a^9 (4 b^4 - 14 b^3 c + 15 b^2 c^2 - 14 b c^3 + 4 c^4) -
3 a^7 (2 b^6 - 6 b^5 c + 6 b^4 c^2 - 7 b^3 c^3 + 6 b^2 c^4 -
6 b c^5 + 2 c^6) -
a^4 (b + c)^3 (4 b^6 - 16 b^5 c + 26 b^4 c^2 - 27 b^3 c^3 +
26 b^2 c^4 - 16 b c^5 + 4 c^6) -
a^3 (b^2 - c^2)^2 (7 b^6 - 17 b^5 c + 11 b^4 c^2 - 6 b^3 c^3 +
11 b^2 c^4 - 17 b c^5 + 7 c^6) +
a^6 (6 b^7 - 4 b^6 c + 7 b^5 c^2 + 4 b^4 c^3 + 4 b^3 c^4 +
7 b^2 c^5 - 4 b c^6 + 6 c^7) +
a^5 (9 b^8 - 20 b^7 c + 3 b^6 c^2 + 3 b^5 c^3 + 6 b^4 c^4 +
3 b^3 c^5 + 3 b^2 c^6 - 20 b c^7 + 9 c^8) : ... : ...)

Exact trilinear coordinates of W' at the triangle ETC (6-9-13):

(-0.115636704440395011, 5.32053318377961059, 0.0105892182632925432)

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*** Remark:

The points O1, O2, O3, O1', O2', O3', the cyclologic centers X(1),
X(1320), X(3065), Z, T, T' and orthologic centers W, W' lies on the
rectangular circum-hyperbola (Feuerbach Hyperbola) through X(21).

Best regards,
Angel Montesdeoca