linetriangle

A linetriangle is a triangle defined by three lines in the plane, with each side lying on one of the lines. The term is not a standard mathematical label, but it is used in some contexts to emphasize that the sides are determined by a line arrangement rather than by a pre-drawn polygon.

Construction and degenerate cases: Let l1, l2, and l3 be three lines in general position, meaning no two are parallel and they do not all meet at a single point. The pairwise intersections A = l2 ∩ l3, B = l1 ∩ l3, and C = l1 ∩ l2 are the vertices of the linetriangle. The region bounded by l1, l2, and l3 is the linetriangle. If two lines are parallel or all three lines pass through a common point, the figure degenerates (the area is zero or no finite triangle is formed).

Computation and properties: If the lines are given in standard form ai x + bi y + ci = 0, one can compute the vertices A, B, C by solving the corresponding pairs of linear equations. The area of the linetriangle is given by 1/2 times the absolute value of the cross product of two edge vectors, equivalently 1/2 |det(B − A, C − A)|. The side lengths depend on the distances between the intersection points and the angles between the lines.

Example: Consider the lines x = 0, y = 0, and x + y = 1. Their intersections are A = (0,0), B = (0,1), and C = (1,0). The linetriangle is the right triangle with area 1/2.

Context: In computational geometry and line-arrangement problems, triangles formed by three lines are common objects of study. The term linetriangle emphasizes its construction from lines rather than from a given set of vertices.