Collinearity in the context of "Triangles"

In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that

• given one set of collinear points ${\displaystyle A,B,C,}$ and another set of collinear points ${\displaystyle a,b,c,}$ then the intersection points ${\displaystyle X,Y,Z}$ of line pairs ${\displaystyle Ab}$ and ${\displaystyle aB,Ac}$ and ${\displaystyle aC,Bc}$ and ${\displaystyle bC}$ are collinear, lying on the Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon ${\displaystyle AbCaBc}$.

It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem" is valid are called pappian planes.

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex; the shortest segment between the base and apex is the height. The area of a triangle equals one-half the product of height and base length.

In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated within a unique flat plane. More generally, four points in three-dimensional Euclidean space determine a solid figure called tetrahedron.

In solid mechanics, shearing forces are unaligned forces acting on one part of a body in a specific direction, and another part of the body in the opposite direction. When the forces are collinear (aligned with each other), they are called tension forces or compression forces. Shear force can also be defined in terms of planes: "If a plane is passed through a body, a force acting along this plane is called a shear force or shearing force."