Triangle

A triangle is a fundamental polygon in geometry.

It consists of three line segments called sides. These segments connect at three points known as vertices. These vertices are zero-dimensional points. The sides, or edges, are one-dimensional line segments. A triangle is considered a plane figure. This means its interior forms a flat, planar region. Any three points define a unique triangle if they do not lie on a single straight line. In three-dimensional space, four points can form a solid called a tetrahedron.

Triangles have specific mathematical properties regarding their internal angles. The sum of the three internal angles always equals a straight angle. This total is exactly 180 degrees or π radians. This specific rule is equivalent to Euclid's parallel postulate. If you know the measure of two angles, you can always find the third. An exterior angle is formed by extending one side. The measure of an exterior angle equals the sum of the two non-adjacent internal angles. Furthermore, the sum of all three exterior angles in any convex polygon is 360 degrees.

Mathematicians classify triangles into distinct types based on side lengths and angles. We name them based on their sides as equilateral, isosceles, or scalene. An equilateral triangle has three equal sides. An isosceles triangle has two equal sides. A scalene triangle has three different side lengths. We also classify them by their angles as acute, right, or obtuse. An acute triangle has all angles less than 90 degrees. A right triangle contains one 90-degree angle. An obtuse triangle has one angle greater than 90 degrees.

The terminology for these classifications is over two thousand years old. Most of these definitions come from Book One of Euclid's Elements. Euclid was a mathematician who laid the groundwork for geometry. The names used today are often direct transliterations of his original Greek. Many are also derived from Latin translations. This ancient system remains the standard for modern geometry. Even the study of trigonometry relies on these classical definitions. Trigonometry uses the sine, cosine, and tangent functions to relate side lengths and angles.

Triangles possess several special points created by intersecting lines. The perpendicular bisectors of the sides meet at the circumcenter. This point is the center of the circumcircle, which passes through all three vertices. The altitudes of a triangle meet at the orthocenter. The angle bisectors meet at the incenter. This incenter is the center of the incircle, which is the largest circle that fits inside the triangle. The medians, which connect vertices to opposite midpoints, meet at the centroid. The centroid is also the geometric barycenter. For a uniform triangular object, this is its center of mass.

These special points often follow very specific geometric patterns. For example, the orthocenter, the centroid, and the circumcenter all lie on a single line. This is known as Euler's line. Additionally, the midpoints of the sides and the feet of the altitudes lie on a nine-point circle. The radius of this nine-point circle is exactly half the radius of the circumcircle. These relationships show how deeply connected the parts of a triangle are. Even simple measurements like area follow strict rules. The area is equal to one-half the product of the base and the height.

Triangles are highly visible in both nature and human engineering. In construction, isosceles triangles appear in gables and pediments. The equilateral triangle is seen on yield signs. The Great Pyramid of Giza uses triangles for its faces. While some assume they are equilateral, they are actually isosceles. Triangles also appear in heraldry, such as on the flags of the Philippines and Saint Lucia. In three dimensions, triangles form the faces of polyhedra. For instance, deltahedra are solids made entirely of equilateral triangles. They even appear in the study of molecular geometry.