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Tangential Triangle


TangentialTriangle

The tangential triangle is the triangle DeltaT_AT_BT_C formed by the lines tangent to the circumcircle of a given triangle DeltaABC at its vertices. It is therefore antipedal triangle of DeltaABC with respect to the circumcenter O. It is also anticevian triangle of DeltaABC with the symmedian point K as the anticevian point (Kimberling 1998, p. 156). Furthermore, the symmedian point K of DeltaABC is the Gergonne point of DeltaT_AT_BT_C.

The tangential triangle is the polar triangle of the circumcircle.

Its trilinear vertex matrix is

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

The side lengths of the tangential triangle are

a^'=(2a^3bc)/(|a^4-(b^2-c^2)^2|)
(2)
b^'=(2ab^3c)/(|b^4-(c^2-a^2)^2|)
(3)
c^'=(2abc^3)/(|c^4-(a^2-b^2)^2|).
(4)

For an acute triangle, the perimeter is given by

 a^'+b^'+c^'=asecA+bsecB+csecC
(5)

and the product of side lengths is

 a^'b^'c^'=1/8abcsec^2Asec^2Bsec^2C.
(6)

The area of the tangential triangle is

 Delta_I=1/2Delta|secAsecBsecC|,
(7)

where Delta is the triangle area of DeltaABC.

The following table gives the centers of the tangential triangle in terms of the centers of the reference triangle that correspond to Kimberling centers X_n.

X_ncenter of tangential triangleX_ncenter of reference triangle
X_2triangle centroidX_(154)X_3-Ceva conjugate of X_6
X_3circumcenterX_(26)circumcenter of the tangential triangle
X_4orthocenterX_(155)eigencenter of orthic triangle
X_5nine-point centerX_(156)X_5-of-tangential-triangle
X_6symmedian pointX_(157)X_6-of-tangential-triangle
X_(30)Euler infinity pointX_(1154)isogonal conjugate of X_(1141)
X_(523)isogonal conjugate of X_(110)X_(1510)Napoleon crossdifference
TangentialTriangleLine

Given a triangle DeltaA_1A_2A_3 and its tangential triangle DeltaT_1T_2T_3, the extensions of the sides of the two triangles intersect in three points L_1, L_2, and L_3, which are collinear (Honsberger 1995).

OrthicTangentialTriangle

The sides of an orthic triangle are parallel to the tangents to the circumcircle at the vertices (Johnson 1929, p. 172). This is equivalent to the statement that each line from a triangle's circumcenter to a vertex is always perpendicular to the corresponding side of the orthic triangle (Honsberger 1995, p. 22), and to the fact that the orthic and tangential triangles are homothetic.


See also

Circumcircle, Contact Triangle, Gergonne Point, Pedal Triangle, Perspective, Tangential Circle, Tangential Quadrilateral

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References

Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 89, 1913.Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 151-153, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Tangential Triangle

Cite this as:

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

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