# Tangential triangle Information

*https://en.wikipedia.org/wiki/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 the reference triangle's
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,^{
[1]}^{:p. 104, p. 242} 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).^{
[2]}^{:p. 447}^{
[1]}^{:p. 102}

The tangential triangle is
homothetic to the
orthic triangle.^{
[1]}^{:p. 98}

A reference triangle and its tangential triangle are in
perspective, and the axis of perspectivity is the
Lemoine axis of the reference triangle. That is, the lines connecting the vertices of the tangential triangle and the corresponding vertices of the reference triangle are
concurrent.^{
[1]}^{:p. 165} The center of perspectivity, where these three lines meet, is the
symmedian point of the 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.^{
[3]}^{:p. 214}

The reference triangle's circumcircle, its
nine-point circle, its
polar circle, and the circumcircle of the tangential triangle are
coaxal.^{
[1]}^{:p. 241}

A right triangle has no tangential triangle, because the tangent lines to its circumcircle at its acute vertices are parallel and thus cannot form the sides of a triangle.

The reference triangle is the Gergonne triangle of the tangential triangle.

## See also

## References

- ^
^{a}^{b}^{c}^{d}^{e}Altshiller-Court, Nathan.*College Geometry*, Dover Publications, 2007 (orig. 1952). **^**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. 1929).