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Anticomplementary Triangle


AnticomplementaryTriangle

The anticomplementary triangle is the triangle DeltaA_1^'A_2^'A_3^' which has a given triangle DeltaA_1A_2A_3 as its medial triangle. It is therefore the anticevian triangle with respect to the triangle centroid G (Kimberling 1998, p. 156), and is in perspective with DeltaABC at G.

It is the polar triangle of the Steiner circumellipse.

Its trilinear vertex matrix is

 [-a^(-1) b^(-1) c^(-1); a^(-1) -b^(-1) c^(-1); a^(-1) b^(-1) -c^(-1)]
(1)

or

 [-bc ac ab; bc -ca ba; cb ca -ab].
(2)

The sides of the anticomplementary triangle are DeltaA_1A_2A_3's exmedians and the vertices are the exmedian points of DeltaA_1A_2A_3.

The circumcircle of the anticomplementary triangle is the anticomplementary circle.

The following table gives the centers of the anticomplementary triangle in terms of the centers of the reference triangle for Kimberling centers X_n with n<=100.

X_ncenter of anticomplementary triangleX_ncenter of reference triangle
X_1incenterX_8Nagel point
X_2triangle centroidX_2triangle centroid
X_3circumcenterX_4orthocenter
X_4orthocenterX_(20)de Longchamps point
X_5nine-point centerX_3circumcenter
X_6symmedian pointX_(69)symmedian point of the anticomplementary triangle
X_7Gergonne pointX_(144)anticomplement of X_7
X_8Nagel pointX_(145)anticomplement of Nagel point
X_9mittenpunktX_7Gergonne point
X_(10)Spieker centerX_1incenter
X_(11)Feuerbach pointX_(100)anticomplement of Feuerbach point
X_(12)(X_1,X_5)-harmonic conjugate of X_(11)X_(2975)internal similitude center(circumcircle, AC-incircle)
X_(13)first Fermat pointX_(616)anticomplement of X_(13)
X_(14)second Fermat pointX_(617)anticomplement of X_(14)
X_(15)first isodynamic pointX_(621)anticomplement of X_(15)
X_(16)second isodynamic pointX_(622)anticomplement of X_(16)
X_(17)first Napoleon pointX_(627)anticomplement of X_(17)
X_(18)second Napoleon pointX_(628)anticomplement of X_(18)
X_(21)Schiffler pointX_(2475)anticomplement of X_(21)
X_(30)Euler infinity pointX_(30)Euler infinity point
X_(25)homothetic center of orthic and tangential trianglesX_(1370)anticomplement of X_(25)
X_(32)third power pointX_(315)isotomic conjugate of X_(66)
X_(37)crosspoint of incenter and triangle centroidX_(75)isotomic conjugate of incenter
X_(39)Brocard midpointX_(76)third Brocard point
X_(40)Bevan pointX_(962)Longuet-Higgins point
X_(44)X_6-line conjugate of X_1X_(320)isotomic conjugate of X_(80)
X_(51)triangle centroid of orthic triangleX_(2979)triangle centroid of dual triangle of X_4
X_(54)Kosnita pointX_(2888)anticomplementary conjugate of X_3
X_(57)isogonal conjugate of X_9X_(329)isotomic conjugate of X_(189)
X_(58)isogonal conjugate of X_(10)X_(1330)anticomplementary conjugate of X_1
X_(61)isogonal conjugate of X_(17)X_(633)anticomplement of X_(61)
X_(62)isogonal conjugate of X_(18)X_(634)anticomplement of X_(62)
X_(69)symmedian point of the anticomplementary triangleX_(193)X_4-Ceva conjugate of X_2
X_(74)X_(74)X_(146)reflection of X_(20) in X_(110)
X_(75)isotomic conjugate of incenterX_(192)X_1-Ceva conjugate of X_2
X_(76)third Brocard pointX_(194)X_6-Ceva conjugate of X_2
X_(81)cevapoint of incenter and symmedian pointX_(2895)anticomplementary conjugate of X_(75)
X_(83)cevapoint of triangle centroid and symmedian pointX_(2896)anticomplementary conjugate of X_(76)
X_(86)cevapoint of incenter and triangle centroidX_(1654)first Hatzipolakis parallelian point
X_(98)Tarry pointX_(147)Tarry point of anticomplementary triangle
X_(99)Steiner pointX_(148)Steiner point of anticomplementary triangle
X_(100)anticomplement of Feuerbach pointX_(149)reflection of X_(20) in X_(104)

The medial triangle DeltaM_1M_2M_3 of a triangle DeltaA_1A_2A_3 is similar to DeltaA_1A_2A_3 and its side lengths are

M_BM_C=2a
(3)
M_CM_A=2b
(4)
M_AM_B=2c.
(5)

This follows immediately by inspecting the construction of the medial triangle and noting that the three vertex triangles and central triangle each have sides of length 2a, 2b, and 2c. Similarly, each of these triangles, including DeltaA^'B^'C^', have area

 Delta^'=4Delta,
(6)

where Delta is the triangle area of DeltaABC.


See also

Anticevian Triangle, Anticomplementary Circle, Exmedian, Exmedian Point, Johnson Circles, Medial Triangle, Triangle Centroid

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References

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Anticomplementary Triangle

Cite this as:

Weisstein, Eric W. "Anticomplementary Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AnticomplementaryTriangle.html

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