Extended side

Trilinear coordinates locate a point in the plane by its relative distances from the extended sides of a reference triangle.

In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.[1]: p.

149 In a triangle, three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.[1]: p.

The excenter (center of the tangent circle) lies at the intersection of six angle bisectors.

Pascal's theorem states that if six arbitrary points are chosen on a conic section (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon.

Each of a triangle's excircles (orange) is tangent to one of the triangle's sides and to the other two extended sides.
An ex-tangential quadrilateral ABCD and its excircle
The intersections of the extended opposite sides of inscribed hexagon ABCDEF lie on the blue Pascal line MNP. The hexagon's extended sides are in gray and red.