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Line at Infinity

The straight line on which all Points at Infinity lie. The line at infinity is given in terms of Trilinear Coordinates by

\begin{displaymath}
a\alpha+b\beta+c\gamma=0,
\end{displaymath}

which follows from the fact that a Real Triangle will have Positive Area, and therefore that

\begin{displaymath}
2\Delta=a\alpha+b\beta+c\gamma>0.
\end{displaymath}

Instead of the three reflected segments concurring for the Isogonal Conjugate of a point $X$ on the Circumcircle of a Triangle, they become parallel (and can be considered to meet at infinity). As $X$ varies around the Circumcircle, $X^{-1}$ varies through a line called the line at infinity. Every line is Perpendicular to the line at infinity.

See also Point at Infinity




© 1996-9 Eric W. Weisstein
1999-05-25