Nagel Triangle and Point
The Nagel triangle of ABC is denoted by the vertices XA, XB and XC that are the three points where the excircles touches the reference triangle ABC and where XA is opposite of A, etc. This triangle XAXBXC is also known as the extouch triangle of ABC. The circumcircle of the extouch triangle XAXBXC is called the Mandart circle. The three lines AXA, BXB and CXC 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.
Read more about this topic: Incircle And Excircles Of A Triangle
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