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K801

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X(2), X(5), X(182), X(524), X(627), X(628), X(641), X(642), X(3413), X(3414), X(7617), X(8149), X(8150), X(11261), X(17748)

imaginary foci of the Brocard inellipse (intersections of the Kiepert hyperbola and the Brocard axis)

K801 is the locus of the circumcenter O(t) of the triangle whose vertices A(t), B(t), C(t) are the apices of the Kiepert triangles when the base angle t varies.

K801 is a nodal cubic with node X(2) and nodal tangents passing through the Fermat points X(13), X(14).

K801 has three real asymptotes : two are the parallels at X(5) to those of the Kiepert hyperbola and the third is the line X(182), X(524).

Note that two opposite values of t correspond to two points O(t), O(-t) collinear with X(182), the center of the Brocard circle.

See Table 32 for other related cubics denoted T(k). In particular, since the centroid of A(t)B(t)C(t) is X(2) for any t, the loci of the orthocenter H(t) and nine point center N(t) of A(t)B(t)C(t) are the anticomplement and complement of K801 respectively.

See also Table 62 where K801 is KWpK(X5) and the analogous K906 = KWpK(X3) which is the anticomplement of K801.
K801a

 

The following table gives the points on K801 and the corresponding values of t and/or cot(t).

t

cot(t)

O(t)

-t

cot(-t)

O(-t)

remarks

0

∞

X(5)

0

∞

X(5)

 

π/2

0

X(524)

-π/2

0

X(524)

 

π/6

√3

X(2)

-π/6

-√3

X(2)

 

π/4

1

X(641)

-π/4

-1

X(642)

 

π/3

1/√3

X(627)

-π/3

-1/√3

X(628)

 

ω

cot(ω)

X(11261)

-ω

- cot(ω)

X182)

see note 1, ω is the Brocard angle

π/2-ω

tan(ω)

X(8150)

-π/2+ω

- tan(ω)

X(8149)

 

t1

cot(t1)

X(3413)

-t1

cot(-t1)

T1

see note 2

t2

cot(t2)

X(3414)

-t2

cot(-t2)

T2

see note 2

 

3 tan(ω)

T

 

- 3 tan(ω)

X(7617)

see note 3

note 1 : X(11261) is the tangential of X(182) on the lines X(2)X(51), X(115)X(3094).

note 2 : cot(t1) = - cot(ω) + √(cot2(ω) - 3), cot(t2) = - cot(ω) - √(cot2(ω) - 3).

The corresponding points T1, T2 have SEARCH numbers 5.33598364143564 and 1.59440300777699 respectively.

note 3 : T lies on the lines X(2)X(187) and X(182)X(7617) with SEARCH = 3.55039924374583.