Triangle · Principia

The figure enclosed by joining three non-collinear points pairwise with segments; the three points are its vertices and the three segments its sides—the simplest polygon, and the most rigid shape in geometry.

Dependencies: Segment, Point–line axiom. A derived definition: three segments joined end to end form the simplest closed polygon.

Definition

Let $A,B,C$ be three noncollinear points. By Point–line axiom, each pair determines a unique line; noncollinear means the third point is not on that line.

Three noncollinear points A,B,C joined by three segments form a triangle.

The figure formed by segments $AB$ , $BC$ , and $CA$ is called triangle $\triangle ABC$ . The three points are its vertices, the segments are its sides, and the angles between pairs of sides are its interior angles.

A triangle is the simplest closed polygon: two segments cannot enclose a region, while three can. It is also rigid—fixing its three side lengths fixes its shape, the basis of triangle congruence.

Remarks

Noncollinearity is essential. If all three points lie on one line, the segments collapse into a degenerate triangle with no area.
Three sides and three angles. These are the triangle's six basic measurements. Its interior angles sum to $180^\circ$ (Triangle interior angles sum to 180°).
Classification by sides and angles. Equal sides give an isosceles triangle; three equal sides give an equilateral triangle. By angles, triangles may be right, obtuse, or acute.
The simplest polygon. More complicated polygons, including quadrilaterals, are often studied by dividing them into triangles with diagonals.