Triangle

• 1. Introduction

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  A triangle holds the distinction of being one of the basic shapes in geometry, built from just three corners and three sides. The corners, called vertices, are zero-dimensional points. The sides connecting them, called edges, are one-dimensional line segments. From this minimal arrangement comes a figure whose three internal angles always sum to a straight angle, 180 degrees or pi radians. The terminology for sorting these shapes is more than two thousand years old, set down in Book One of Euclid's Elements. So how does a figure this simple generate centers, circles, and lines named after mathematicians across centuries? Why do engineers reach for it when they want something that will not collapse? And what happens to its tidy rules when you draw it on a sphere or a saddle instead of a flat page?

• 2. A Family Of Names

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  Euclid's Greek terms, transliterated directly or translated into Latin, still govern how triangles are named today. A triangle whose sides are all the same length is equilateral. One with two sides of equal length is isosceles, and one with three different lengths is scalene. The angles supply a second naming system that runs in parallel. A triangle containing a right angle is a right triangle. When every angle stays below that, the triangle is acute, and when one angle exceeds it, the triangle is obtuse. These definitions reach back at least to Euclid himself, who first separated the categories in Book One of his Elements.

• 3. Triangles In The World

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  The yield sign carries an equilateral triangle that any driver would recognize. Gables and pediments in man-made construction take the form of isosceles triangles, and the flag of Saint Lucia and the flag of the Philippines both place the shape into heraldic symbols. The faces of the Great Pyramid of Giza are sometimes called equilateral, though more accurate measurements show them to be isosceles instead. Triangles climb into three dimensions as well, covering the faces of solids called polyhedra. When a polyhedron has all equilateral triangles for its faces, it is known as a deltahedron. Antiprisms carry alternating triangles along their sides, while pyramids and bipyramids use triangles for their lateral faces. The Kleetope replaces each face of a polyhedron with a pyramid, so its faces become triangles too. Push into higher dimensions and the triangle generalizes into the simplex, the seed of the simplicial polytopes.

• 4. Centers And Circles

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  Ceva's theorem gives a criterion for deciding when three lines drawn through a triangle meet at a single point, and that idea unlocks a crowd of special points hiding inside the figure. The three perpendicular bisectors of the sides converge at the circumcenter, the center of the circumcircle that passes through all three vertices. Where the circumcenter sits reveals the triangle's character. Inside means acute, outside means obtuse, and resting on a side means a right angle, by Thales' theorem. The three altitudes, each running from a vertex perpendicular to the opposite side, meet at the orthocenter, which lies inside the triangle exactly when the triangle is acute. The three angle bisectors converge at the incenter, the center of the incircle, the largest circle that fits inside and touches all three sides. Three excircles lie outside, each touching one side and the extensions of the other two. The midpoints of the sides and the feet of the altitudes share a single circle, the nine-point circle, whose radius is half that of the circumcircle. The orthocenter, the center of the nine-point circle, the centroid, and the circumcenter all sit on one line called Euler's line. The three medians, each joining a vertex to the midpoint of the opposite side, meet at the centroid, which cuts every median in a 2-1 ratio. Cut a triangle from a thin sheet of uniform density and it balances on that very point.

• 5. The Logic Of Angles

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  The 180-degree rule for a triangle's interior angles is equivalent to Euclid's parallel postulate, so the two stand or fall together. Knowing two angles immediately fixes the third. An exterior angle, supplementary to its interior partner, equals the sum of the two interior angles not adjacent to it, a result called the exterior angle theorem. Add up the three exterior angles and you always reach 360 degrees, a total that holds for any convex polygon regardless of how many sides it has. Specifying the three angles does not pin down the size, so infinitely many triangles can share the same set of angles. A degenerate triangle, whose vertices fall on a single line, carries internal angles of 0 degrees and 180 degrees, and whether it counts as a triangle at all is a matter of convention. The ratios between sides of a right triangle give rise to the sine and cosine functions, the tools that let the law of sines and the law of cosines recover an unknown side or angle in a scalene triangle.

• 6. Sameness And Proof

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  Two triangles are similar when every angle of one matches the corresponding angle of the other, which forces their sides into the same proportion. Matching just two pairs of angles is enough to prove it. So is a single pair of sides in proportion with their included angles equal, or all three pairs of sides sharing one proportion. Congruence is the stronger relation, demanding the same size and shape, a total of six matching equalities between angles and sides. Three of those equalities usually suffice. The SAS postulate pairs two sides and the included angle, while SSS compares all three sides directly. ASA matches two angles and the side between them, the principle behind surveying by triangulation, and AAS uses two angles with a non-included side.

• 7. Measuring The Interior

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  Area in the Euclidean plane is measured against a square of side length one, which is assigned an area of 1. The oldest and simplest method for a triangle takes half the product of a base and its corresponding altitude, a result proven by cutting a triangle and its copy into pieces that rearrange into a rectangle. Heron's formula, named after Heron of Alexandria, reaches the same area from the three side lengths alone by way of the semiperimeter. Because affine transformations preserve the ratios between areas, relative areas can be defined without any notion of distance, an approach laid out in Book One of Euclid's Elements. The shoelace formula computes a triangle's oriented area straight from the Cartesian coordinates of its vertices using a matrix determinant. The triangle inequality sets the boundary on which side lengths are even possible. The sum of any two sides must be greater than or equal to the third, with equality only in the degenerate case of collinear vertices.

• 8. Strength And Strange Spaces

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  Press on a corner of a rectangle and it folds into a parallelogram, but a triangle refuses. Fixing the lengths of all three sides locks the angles in place, so each side supports the other two and the shape holds unless something bends, stretches, or breaks. This is why structural quadrilaterals are braced with a diagonal that splits them into two rigid triangles. Packed in a tessellation, triangles yield to hexagons under compression, which is part of why hexagonal forms appear so often in nature. Yet triangles keep the upper hand for cantilevering, the reason engineering relies on tetrahedral trusses. The flat-page rules loosen entirely off the plane. On a negatively curved saddle surface, a hyperbolic triangle's angles sum to less than 180 degrees, while on a sphere a spherical triangle exceeds it. A triangle can be drawn on a sphere with three right angles, each 90 degrees, adding up to 270. By Girard's theorem, the angle excess of a spherical triangle measures the fraction of the sphere's area it encloses. The same three-sided form even seeds fractals, repeating into the Sierpinski gasket and the Koch snowflake.