Tangential triangle

In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices. Thus the incircle of the tangential triangle coincides with the circumcircle of the reference triangle.

The circumcenter of the tangential triangle is on the reference triangle's Euler line, as is the center of similitude of the tangential triangle and the orthic triangle (whose vertices are at the feet of the altitudes of the reference triangle).^[1]^{: p. 447}

The tangential triangle is homothetic to the orthic triangle.

A reference triangle and its tangential triangle are in perspective, and the axis of perspectivity is the Lemoine axis of the reference triangle.

The tangent lines containing the sides of the tangential triangle are called the exsymmedians of the reference triangle. Any two of these are concurrent with the third symmedian of the reference triangle.^[2]^{: p. 214}

References

^ Smith, Geoff, and Leversha, Gerry, "Euler and triangle geometry", Mathematical Gazette 91, November 2007, 436–452.
^ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).

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[SL-1] Smith, Geoff, and Leversha, Gerry, "Euler and triangle geometry", Mathematical Gazette 91, November 2007, 436–452.

[Johnson-2] Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).

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