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K1253

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X(55), X(57), X(165), X(365), X(1615), X(8830), X(8831), X(8832), X(8833), X(8917), X(11051), X(46376), X(46377), X(46378), X(46379)

X165) Ceva conjugate of X(11051)

A1B1C1 : cevian triangle of X(165)

A2B2C2 : circumcevian triangle of X(55) i.e. 4th mixtilinear triangle, see ETC, preambles before X(7955) and X(8782)

A3B3C3 : anticevian triangle of X(365)

A4, B4, C4 : points on the antiorthic axis and on the cevian lines of X(57)

A5B5C5 : cevian triangle of X(57) wrt A2B2C2

Geometric properties :

The 4th mixtilinear triangle and the cevian (resp. anticevian) triangle of M are perspective if and only if M lies on pK(X1253, X9) (resp. K1403 = pK(X31, X1)). In both cases, the locus of the perspector N is K1253. See table below where other mixtilinear triangles are considered.

See Q174 for another locus property.

K1253 is a circumcevian pK, see CL072.

K1253 is the isogonal transform of K202 = pK(X1, X144).

Notes :

X(55) is a point of inflexion with inflexional tangent passing through X(6).

The tangents at A, B, C, X(165) concur at the isopivot X(11051).

The tangents at A1, B1, C1, X(11051) concur at the third pivot, the X165) Ceva conjugate of X(11051).

The tangents at A2, B2, C2, X(57) concur at X(1615).

The tangents at A3, B3, C3, X(365) concur at X(165). The polar conic of X(165) is a diagonal rectangular hyperbola passing through X(i) for these i : 1, 43, 165, 170, 365, 846, 1051, 1282, 2108, 2536, 2537, 2939, 2944, 2947, 2954, etc.

The tangents at A4, B4, C4, X(165) concur at X(55) since the antiorthic axis is the harmonic polar of the point of inflexion X(6).

The tangents at A5, B5, C5, X(1615) concur at the barycentric quotient X(31) รท X(8830).

Any two of the six triangles above are perspective at a point on the cubic.

***

See also preambles just before X(8782) and X(8856) in ETC for other triangles.

mixtilinear triangles

cevian locus of M

cevian/anticevian locus of N

anticevian locus of M

1st

pK(X8828, X8829)

pK(X198, X16977)

pK(X198, X57)

3rd

pK(X7366, X269)

pK(X604, X1420)

isogonal of K1077

4th

pK(X1253, X9)

K1253

isogonal of K363

5th

K365

K201

K1077

6th

K202

pK(X6, X2951)

K351

7th

K202

pK(X269, X15913)

pK(X269, X7)

Remark 1 : the 2nd mixtilinear triangle gives the union of the internal bisectors of ABC for each locus.

Remark 2 : the isogonal transforms of K1077 and K363 are pK(X604, X57) and pK(X31, X1) respectively.