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X(1), X(2), X(3), X(4), X(6), X(7), X(354), X(672), X(955), X(991), X(5728), X(14547), X(14548), X(14549)

other points below

K382 is the only (undecomposed) cubic passing through I, G, O, H, K and Ge (Gergonne point) i.e. six centers of the ETC Top 10. See also K696 and Table 30.

The following table gives alignments and other points on the cubic.

Xi

Xj

Xk / pt

notes/1st baryc.

Search

1

2

1

G is the tangential of I

*

1

3

354

*

1

4

14547

 

*

1

6

5728

 

*

1

7

991

*

1

672

a*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4 + a^4*c + a^3*b*c - 2*a^2*b^2*c - a*b^3*c + b^4*c - a^3*c^2 - 2*a^2*b*c^2 - 4*a*b^2*c^2 - b^3*c^2 - a^2*c^3 - a*b*c^3 - b^2*c^3 + a*c^4 + b*c^4)/(2*b*c + a*b + a*c - a^2)

1.5149423271

1

995

ugly

0.39759327634

2

3

4

*

2

6

14548

 

*

2

7

672

*

2

354

a(a^2+b^2+c^2-2*a*b-2*a*c)/(-a^2+a*b+a*c+2*b*c)

-0.26879818663

2

955

5728

 

*

2

991

1/(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c - a^2*b*c + a*b^2*c + b^3*c - a^2*c^2 + a*b*c^2 - 2*b^2*c^2 + a*c^3 + b*c^3)

6.1661934751

3

6

991

*

3

7

OGe

(3*a^2*b-2*a*b^2-b^3+3*a^2*c+4*a*b*c+b^2*c-2*a*c^2+b*c^2-c^3)/(a^2*b-b^3+a^2*c+2*a*b*c+b^2*c+b*c^2-c^3)

1.8270354007

3

672

a^2(a*b+a*c-b^2-c^2)/(-a^3+a^2*b-a*b^2+b^3+a^2*c-2*a*b*c-b^2*c-a*c^2-b*c^2+c^3)

-12.113065373

3

955

ugly

1.794898451968

4

6

HK

ugly

-31.019596589

4

7

5728

 

*

4

354

ugly

1.533666290999

4

672

ugly

-4.8466314073

4

955

ugly

1.198832957781

4

991

14549

 

*

6

7

ugly

2.4103172619

6

354

a*(a^5 - a^4*b - a*b^4 + b^5 - a^4*c - 2*a^3*b*c + 2*a^2*b^2*c + 2*a*b^3*c - b^4*c + 2*a^2*b*c^2 + 6*a*b^2*c^2 + 2*a*b*c^3 - a*c^4 - b*c^4 + c^5)/(a^2 + b^2 + c^2 - 2*a*b - 2*a*c)

18.74219943685

6

672

14547

 

*

6

955

ugly

1.769623144772

7

354

14548

 

*

7

955

ugly

-0.4068239559082

354

672

14549

 

*

354

955

HK

ugly

-31.019596589

354

991

a^2*(a^2 - b^2 - 4*b*c - c^2)/(-a^5 + 2*a^4*b - 2*a^2*b^3 + a*b^4 + 2*a^4*c + 2*a^3*b*c - 5*a^2*b^2*c + b^4*c - 5*a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - 2*a^2*c^3 - b^2*c^3 + a*c^4 + b*c^4)

1.658226689587

672

955

ugly

1.664615036028

672

991

ugly

-0.880651271322

955

991

14547

 

*