Definition of a Triangle in Argumentative Geometry
Definition
Given three distinct points $A$, $B$, $C$, the three line segments $\overline{AB}$, $\overline{BC}$, $\overline{CA}$ constitute the triangle $\triangle{ABC}$, which has sides $\overline{AB}$, $\overline{BC}$, $\overline{CA}$ and vertices $A$, $B$, $C$.
Explanation
This definition is taken from the Birkhoff axiom system, and it is a convenient formulation because, compared with the Euclidean axiom system or the Hilbert axiom system, it is able to refer to sides and vertices simultaneously.
Fundamentally, since every polygon can be decomposed into triangles by connecting vertices, any complex figure can be reduced to studying a single triangle; there is no need to study other shapes separately. However, for quadrilaterals one must study up to trapezoids, because the concept of parallelism does not exist within triangles.