Incircle and Excircles of A Triangle - Nagel Triangle and Point

Nagel Triangle and Point

The Nagel triangle of ABC is denoted by the vertices X_A, X_B and X_C that are the three points where the excircles touches the reference triangle ABC and where X_A is opposite of A, etc. This triangle X_AX_BX_C is also known as the extouch triangle of ABC. The circumcircle of the extouch triangle X_AX_BX_C is called the Mandart circle. The three lines AX_A, BX_B and CX_C intersect in a single point, the triangle's Nagel point Na - X(8).

Trilinear coordinates for the vertices of the intouch triangle are given by

Trilinear coordinates for the vertices of the extouch triangle are given by

Trilinear coordinates for the vertices of the incentral triangle are given by

Trilinear coordinates for the vertices of the excentral triangle are given by

Trilinear coordinates for the Gergonne point are given by

,

or, equivalently, by the Law of Sines,

.

Trilinear coordinates for the Nagel point are given by

,

or, equivalently, by the Law of Sines,

.

It is the isotomic conjugate of the Gergonne point.