TOPICS
Search

Anticomplement


The anticomplement of a point P in a reference triangle DeltaABC is a point P^' satisfying the vector equation

 P^'G^->=2GP^->,
(1)

where G is the triangle centroid of DeltaABC (Kimberling 1998, p. 150).

The anticomplement of a point with center function alpha:beta:gamma is therefore given by the point with trilinears

 (-aalpha+bbeta+cgamma)/a:(aalpha-bbeta+cgamma)/b:(aalpha+bbeta-cgamma)/c.
(2)

The anticomplement of a line

 lalpha+mbeta+ngamma=0
(3)

is given by the line

 a^2(cm+bn)alpha+b^2(cl+an)beta+c^2(bl+am)gamma=0.
(4)

The following table summarizes the anticomplements of a number of named lines, including their Kimberling line and center designations.

The following table summarizes the anticomplements of a number of named circles.

The following table lists some points and their anticomplements using Kimberling center designations.

PP^'
X_1X_8
X_2X_2
X_3X_4
X_4X_(20)
X_5X_3
X_6X_(69)
X_7X_(144)
X_8X_(145)
X_9X_7
X_(10)X_1
X_(11)X_(100)
X_(12)
X_(13)X_(616)
X_(14)X_(617)
X_(15)X_(621)
X_(16)X_(622)
X_(17)X_(627)
X_(18)X_(628)
X_(19)
X_(20)
X_(21)X_(2475)
X_(25)X_(1370)
X_(32)X_(315)
X_(37)X_(75)
X_(39)X_(76)
X_(40)X_(962)
X_(44)X_(320)
X_(57)X_(329)
X_(58)X_(1330)
X_(61)X_(633)
X_(62)X_(634)
X_(69)X_(193)
X_(74)X_(146)
X_(75)X_(192)
X_(76)X_(194)
X_(86)X_(1654)
X_(113)X_(74)
X_(114)X_(98)
X_(115)X_(99)
X_(116)X_(101)
X_(117)X_(102)
X_(118)X_(103)
X_(119)X_(104)
X_(120)X_(105)
X_(121)X_(106)
X_(122)X_(107)
X_(123)X_(108)
X_(124)X_(109)
X_(125)X_(110)
X_(126)X_(111)
X_(127)X_(112)
X_(128)X_(1141)
X_(129)X_(1298)
X_(130)X_(1303)
X_(131)X_(1300)
X_(132)X_(1297)
X_(133)X_(1294)
X_(136)X_(925)
X_(137)X_(930)
X_(140)X_5
X_(141)X_6
X_(142)X_9
X_(618)X_(13)
X_(619)X_(14)
X_(623)X_(15)
X_(624)X_(16)
X_(629)X_(17)
X_(630)X_(18)
X_(1125)X_(10)

See also

Complement

Explore with Wolfram|Alpha

References

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

Referenced on Wolfram|Alpha

Anticomplement

Cite this as:

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

Subject classifications