Proving the Three-Sided Nature of Triangles in Euclidean Geometry

A triangle is defined as that three-sided figure. This fundamental property seems so intuitive that one might ask, what needs to be proven? After all, just as it's almost trivial to assert that 'rain is rain' or that 'a cow is a cow,' one might wonder if such basic geometric truths are not self-evident. However, within Euclidean geometry, we delve into the mathematical rigor that makes each concept inherently precise and understandable.

Fellow Quoran Professor David Joyce's comprehensive reference, the online Euclid's Elements, offers valuable insights into this topic. Interestingly, according to Dr. Joyce's version, triangles are not explicitly defined as three-sided figures, though trilateral figures (three-sided figures) are. Triangles are then classified into various types: equilateral, isosceles, and scalene, alongside acute, right, and obtuse triangles. Euclid does not explicitly state that trilaterals and triangles are the same, but it is clear that all triangles are necessarily trilaterals.

Defining Rectilinear Figures and Triangles

Let's examine Euclid’s definitions from Book I for a better understanding:

Definition 19

Rectilinear figures are those which are contained by straight lines. Trilateral figures are those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

Definition 20

Of trilateral figures, an equilateral triangle has its three sides equal; an isosceles triangle has two of its sides alone equal, and a scalene triangle has its three sides unequal.

Definition 21

Further of trilateral figures, a right-angled triangle has a right angle; an obtuse-angled triangle has an obtuse angle; and an acute-angled triangle has its three angles acute.

The point is clear: a triangle is defined to have three sides. However, as is common with Euclid, the definitions are not as precise and unambiguous as we might prefer.

A Proof of a Triangle Having Three Sides

If we need a more rigorous proof, it can be structured as follows:

According to Definition 20, it states that all triangles are trilaterals. Definition 19 defines a trilateral figure as one contained by three straight lines, which, in Euclidean geometry, are understood to mean line segments. Thus, from these definitions, it follows that a triangle, being a trilateral figure, is necessarily contained by three straight line segments, proving that a triangle has three sides.

This meticulous proof from Euclidean geometry confirms the intuitive idea of a triangle's fixed nature. Despite the historical context of Euclid's work, the principle remains a cornerstone of modern geometry, emphasizing the importance of precise definitions and logical deductions.

In summary, while the concept might seem trivial, the rigorous proofs in Euclidean geometry are essential for understanding and applying these fundamental concepts. This exploration not only enhances our understanding but also provides a solid foundation for more complex geometric and mathematical theories.