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

$$a\alpha+b\beta+c\gamma=0,$$

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

$$2\Delta=a\alpha+b\beta+c\gamma>0.$$

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