Medial Triangle


The triangle DeltaM_AM_BM_C formed by joining the midpoints of the sides of a triangle DeltaABC. The medial triangle is sometimes also called the auxiliary triangle (Dixon 1991).

The medial triangle is the Cevian triangle of the triangle centroid G and the pedal triangle of the circumcenter O (Kimberling 1998, p. 155). It is also the cyclocevian triangle of the orthocenter H.

The medial triangle is the polar triangle of the Steiner inellipse.

Its trilinear vertex matrix is

 [0 b^(-1) c^(-1); a^(-1) 0 c^(-1); a^(-1) b^(-1) 0]


 [0 ac ab; bc 0 ab; bc ac 0].

The medial triangle DeltaM_AM_BM_C of a triangle DeltaABC is similar to DeltaABC and its side lengths are


This follows immediately by inspecting the construction of the medial triangle and noting that the three vertex triangles and medial triangle each have sides of length a/2, b/2, and c/2. Similarly, each of these triangles, including DeltaM_AM_BM_C, have area


where Delta is the triangle area of DeltaABC.

The incircle of the medial triangle is called the Spieker circle, and its incenter is called the Spieker center. The circumcircle of the medial triangle is the nine-point circle.

Given a reference triangle DeltaABC, let the angle bisectors of A and B cut the side (or extended side) of the medial triangle DeltaM_AM_B at I_A and I_B. Then CI_A is perpendicular to the angle bisector of A and CI_B is perpendicular to the angle bisector of B. Similarly, by taking pairs of angle bisectors in turn, perpendiculars can be dropped from A and B to their respective intersections with the other sides of the medial triangle (Carding 2006; F. M. Jackson, pers. comm., Aug. 5, 2006).

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

X_ncenter of medial triangleX_ncenter of reference triangle
X_1incenterX_(10)Spieker center
X_2triangle centroidX_2triangle centroid
X_3circumcenterX_5nine-point center
X_5nine-point centerX_(140)midpoint of X_3 and X_5
X_6symmedian pointX_(141)complement of symmedian point
X_7Gergonne pointX_9mittenpunkt
X_8Nagel pointX_1incenter
X_9mittenpunktX_(142)complement of mittenpunkt
X_(10)Spieker centerX_(1125)complement of X_(10)
X_(11)Feuerbach pointX_(3035)complement of X_(11)
X_(13)first Fermat pointX_(618)complement of X_(13)
X_(14)second Fermat pointX_(619)complement of X_(14)
X_(15)first isodynamic pointX_(623)complement of X_(15)
X_(16)second isodynamic pointX_(624)complement of X_(16)
X_(17)first Napoleon pointX_(629)complement of X_(17)
X_(18)second Napoleon pointX_(630)complement of X_(18)
X_(20)de Longchamps pointX_4orthocenter
X_(21)Schiffler pointX_(442)complement of Schiffler point
X_(22)Exeter pointX_(427)complement of X_(22)
X_(23)far-out pointX_(858)complement of X_(23)
X_(25)homothetic center of orthic and tangential trianglesX_(1368)complementary conjugate of X_6
X_(27)Cevapoint of orthocenter and Clawson centerX_(440)complement of X_(27)
X_(30)Euler infinity pointX_(30)Euler infinity point
X_(31)second power pointX_(2887)complementary conjugate of X_(37)
X_(32)third power pointX_(626)complement of X_(32)
X_(38)crosspoint of X_1 and X_(75)X_(1215)isogonal conjugate of X_(1178)
X_(40)Bevan pointX_(946)midpoint of X_1 and X_4
X_(52)orthocenter of orthic triangleX_(1216)isogonal conjugate of X_(1179)
X_(54)Kosnita pointX_(1209)isogonal conjugate of X_(1166)
X_(55)internal similitude center of circumcircle and incircleX_(2886)complementary conjugate of X_9
X_(56)external similitude center of circumcircle and incircleX_(1329)complementary conjugate of X_1
X_(61)isogonal conjugate of X_(17)X_(635)complement of X_(61)
X_(62)isogonal conjugate of X_(18)X_(636)complement of X_(62)
X_(63)isogonal conjugate of X_(19)X_(226)X_7-Ceva conjugate of X_(65)
X_(64)isogonal conjugate of X_(20)X_(2883)complementary conjugate of X_4
X_(65)orthocenter of the intouch triangleX_(960)intersection of lines X_1X_6 and X_5X_(10)
X_(66)isogonal conjugate of X_(22)X_(206)X_2-Ceva conjugate of X_(32)
X_(68)Prasolov pointX_(1147)isogonal conjugate of X_(847)
X_(69)symmedian point of the anticomplementary triangleX_6symmedian point
X_(72)isogonal conjugate of X_(28)X_(942)inverse-in-incircle of X_(36)
X_(74)X_(74)X_(113)Jerabek antipode
X_(75)isotomic conjugate of incenterX_(37)crosspoint of incenter and triangle centroid
X_(76)third Brocard pointX_(39)Brocard midpoint
X_(78)isogonal conjugate of X_(34)X_(1210)isogonal conjugate of X_(1167)
X_(80)reflection of incenter in Feuerbach pointX_(214)X_2-Ceva conjugate of X_(44)
X_(81)Cevapoint of incenter and symmedian pointX_(1211)isogonal conjugate of X_(1169)
X_(85)isotomic conjugate of X_9X_(1212)isogonal conjugate of X_(1170)
X_(86)Cevapoint of incenter and triangle centroidX_(1213)isogonal conjugate of X_(1171)
X_(92)Cevapoint of incenter and Clawson pointX_(1214)isogonal conjugate of X_(1172)
X_(95)Cevapoint of triangle centroid and circumcenterX_(233)X_2-Ceva conjugate of X_(140)
X_(98)Tarry pointX_(114)Kiepert antipode
X_(99)Steiner pointX_(115)center of Kiepert hyperbola
X_(100)anticomplement of Feuerbach pointX_(11)Feuerbach point

See also

Anticomplementary Triangle, Circum-Medial Triangle, Cleavance Center, Cleaver, Median Triangle, Nine-Point Circle, Spieker Center, Spieker Circle, Steiner Inellipse, Triangle Median

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Carding, M. "Culture Shock for Mathematics and Science." Math. Today 42, 129-131, Aug. 2006.Coxeter, H. S. M. and Greitzer, S. L. "The Medial Triangle and Euler Line." §1.7 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 18-20, 1967.Dixon, R. Mathographics. New York: Dover, p. 56, 1991.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Medial Triangle

Cite this as:

Weisstein, Eric W. "Medial Triangle." From MathWorld--A Wolfram Web Resource.

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