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GeezerVison is showing mathworld.wolfram.com — Extouch Triangle -- from Wolfram MathWorld

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Extouch Triangle

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ExtouchTriangle

The extouch triangle DeltaT_1T_2T_3 is the triangle formed by the points of tangency of a triangle DeltaA_1A_2A_3 with its excircles J_1, J_2, and J_3. The points T_1, T_2, and T_3 can also be constructed as the points which bisect the perimeter of DeltaA_1A_2A_3 starting at A_1, A_2, and A_3.

It is the Cevian triangle of the Nagel point (Kimberling 1998, p. 158), the pedal triangle of the Bevan point X_(40), and the cyclocevian triangle of X_(189).

It is the polar triangle of the Mandart inellipse.

Equivalent trilinear vertex matrices for the extouch triangle are

V=[0 (a-b+c)/b (a+b-c)/c; (-a+b+c)/a 0 (a+b-c)/c; (-a+b+c)/a (a-b+c)/b 0]
(1)
V=[0 csc^2(1/2B) csc^2(1/2C); csc^2(1/2A) 0 csc^2(1/2C); csc^2(1/2A) csc^2(1/2B) 0].
(2)

The lengths of the sides of the extouch triangle are

a^'=sqrt(a^2-(4Delta^2)/(bc))
(3)
=sqrt(a^2-bcsin^2A)
(4)
b^'=sqrt(b^2-(4Delta^2)/(ca))
(5)
=sqrt(b^2-casin^2B)
(6)
c^'=sqrt(c^2-(4Delta^2)/(ab))
(7)
=sqrt(c^2-absin^2C),
(8)

where Delta is the triangle area.

The extouch triangle has triangle area

Delta_T=((a+b-c)(a-b+c)(-a+b+c))/(4abc)Delta
(9)
=(2r^2s)/(abc)Delta,
(10)

where Delta, r, and s are the triangle area, inradius, and semiperimeter, respectively, of the original triangle DeltaA_1A_2A_3. This is the same area as the contact triangle.

The circumcircle of the extouch triangle is known as the Mandart circle.

The following table gives all centers of the extouch triangle in terms of the centers of the reference triangle that are Kimberling centers X_n.

X_ncenter of extouch triangleX_ncenter of reference triangle
X_2triangle centroidX_(210)X_(10)-Ceva conjugate of X_(37)
X_3circumcenterX_(1158)circumcenter of extouch triangle

See also

Cevian Triangle, Excenter, Excircles, Mandart Circle, Mandart Inellipse, Nagel Point

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References

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

Referenced on Wolfram|Alpha

Extouch Triangle

Cite this as:

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

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