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Extangents Triangle


ExtangentsTriangle

The triangle T that is externally tangent to the excircles and forms their triangular hull is called the extangents triangle (Kimberling 1998, p. 162). It is homothetic to the orthic triangle, and the homothetic center is known as the Clawson point.

The extangents triangle has trilinear vertex matrix

 [-(x+1) x+z x+y; y+z -(y+1) y+x; z+y z+x -(z+1)],
(1)

where x=cosA, y=cosB, z=cosC, or equivalently,

 [-a/((a+b-c)(a-b+c)) (a+c)/((a-b+c)(a+b+c)) (a+b)/((a+b-c)(a+b+c)); (b+c)/((-a+b+c)(a+b+c)) -b/((a+b-c)(-a+b+c)) (a+b)/((a+b-c)(a+b+c)); (b+c)/((-a+b+c)(a+b+c)) (a+c)/((a-b+c)(a+b+c)) -c/((-a+b+c)(a-b+c))].
(2)

It has area

 Delta^'=([(a^3+b^3+c^3)-(a+b)(b+c)(c+a)]^2secAsecBsecC)/(8a^2b^2c^2)Delta,
(3)

where Delta is the area of DeltaABC.

The circumcircle of the extangents triangle is the extangents circle.

ExcentralTriangleTangent

Its incenter I_T coincides with the circumcenter C_J of triangle DeltaJ_1J_2J_3, where J_i are the excenters of A. The inradius r_T of the incircle of T is

 r_T=2R+r=1/2(r+r_1+r_2+r_3),
(4)

where R is the circumradius of A, r is the inradius, and r_i are the exradii (Johnson 1929, p. 192).


See also

Clawson Point, Excircles, Extangents Circle, Yff Central Triangle

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References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Extangents Triangle

Cite this as:

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

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