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Triangle

A triangle is a polygon with three sides and three vertices, one of the fundamental shapes in Euclidean geometry that forms a closed plane figure.^[1]^[2] It is always convex and consists of three line segments connected end-to-end, with the points of connection known as vertices.^[1] The sum of its interior angles is exactly 180 degrees, and the sides must satisfy the triangle inequality theorem, where the sum of any two sides exceeds the length of the third.^[1]^[2] Triangles are classified into various types based on their side lengths and angle measures, which determine their properties and applications.^[1] By sides, they include equilateral (all sides equal), isosceles (two sides equal), and scalene (all sides of different lengths).^[1]^[2] By angles, triangles are categorized as acute (all angles less than 90 degrees), right (one angle exactly 90 degrees), obtuse (one angle greater than 90 degrees), or equiangular (all angles equal, which coincides with equilateral).^[1]^[2] Key properties include the altitude (perpendicular distance from a vertex to the opposite side), median (line to the midpoint of the opposite side), and the fact that the longest side is opposite the largest angle.^[2] The area can be computed using formulas such as (1/2) × base × height or Heron's formula involving the semiperimeter.^[1]^[2] Triangles serve as building blocks for more complex polygons and are essential in fields like architecture, engineering, and physics due to their structural stability and simplicity.^[1]

Fundamentals

Definition and Terminology

In Euclidean plane geometry, a triangle is defined as a two-dimensional figure bounded by three straight line segments that intersect pairwise, forming three vertices and enclosing a finite region without self-intersections.^[3] It consists of three edges, known as sides, connecting the vertices, and three interior angles at those vertices. The sum of the interior angles of any triangle is exactly 180 degrees, or two right angles, as established in classical geometry. The vertices of a triangle are typically labeled with capital letters, such as A, B, and C, while the sides opposite these angles are denoted by lowercase letters a, b, and c, respectively—a standard convention where side a lies opposite angle A, side b opposite angle B, and side c opposite angle C.^[4] Interior angles are the angles inside the triangle formed by the sides, whereas exterior angles are formed by extending one side beyond a vertex and measuring the angle adjacent to the interior angle at that vertex; each exterior angle is supplementary to its corresponding interior angle, summing to 180 degrees. Triangles in the plane also have an orientation, determined by the clockwise or counterclockwise ordering of their vertices when traversed along the boundary. The term "triangle" originates from the Latin word triangulus, meaning "three-cornered," derived from tri- (three) and angulus (angle or corner).^[5] The earliest known studies of triangles appear in ancient Egyptian and Babylonian mathematics during the second millennium BCE, where they were applied practically in land surveying, pyramid construction, and calculations involving right angles using ratios like the 3-4-5 triple.^[6]^[7]

Classification by Sides and Angles

Triangles are classified based on the relative lengths of their sides into three primary categories: equilateral, isosceles, and scalene. An equilateral triangle has all three sides of equal length, resulting in a high degree of symmetry.^[8] An isosceles triangle features exactly two sides of equal length, with the unequal side known as the base. A scalene triangle, in contrast, has three sides of different lengths, lacking any symmetry in side measures.^[9] Classification by angles divides triangles into acute, right, and obtuse types, depending on the measures of their interior angles. An acute triangle contains three angles each measuring less than 90 degrees.^[10] A right triangle includes exactly one angle of 90 degrees, with the other two angles being acute.^[11] An obtuse triangle has one angle greater than 90 degrees, while the remaining two angles are acute.^[12] This angular classification is possible because the sum of the interior angles in any triangle equals 180 degrees.^[8] Triangles can also be described by combining side and angle classifications to specify both attributes. For instance, an equilateral triangle is always acute, as its equal angles each measure 60 degrees.^[13] A right-isosceles triangle has two equal sides and a right angle, with the other two angles each measuring 45 degrees.^[9] Other combinations, such as an obtuse scalene triangle, feature unequal sides and one obtuse angle.^[11] A degenerate triangle occurs when the three vertices are collinear, collapsing the figure into a straight line segment with zero area, rather than forming a proper triangular region.^[14] This configuration violates the standard triangle inequality in a strict sense, as the points do not enclose a two-dimensional space.^[15]

Geometric Elements

Sides, Angles, and Perimeter

In a triangle ABC, the sides are conventionally denoted by lowercase letters a, b, and c, where a is the length of the side opposite angle A, b opposite angle B, and c opposite angle C.^[1] This standard notation facilitates the expression of relationships between sides and angles. The perimeter P of the triangle is simply the sum of these side lengths:

P = a + b + c.

The semiperimeter s, often used in further geometric formulas, is half of this value: s = \frac{1}{2}(a + b + c).^[1] The three interior angles A, B, and C of any triangle in Euclidean geometry sum to exactly $180^\circ or \pi radians.^[1] This angle sum theorem can be proven by drawing an auxiliary line parallel to one side of the triangle, invoking properties of alternate interior angles formed with transversals, which equate the interior angles to half the sum of straight angles along the parallel line.^[16] An important corollary is the exterior angle theorem: when one side of the triangle is extended, the exterior angle formed is equal to the sum of the two non-adjacent (remote) interior angles.^[17] This follows from the angle sum theorem and the fact that adjacent interior and exterior angles are supplementary (summing to $180^\circ).^[17] A key relation connecting sides and angles is the law of sines, which states that

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,

where R is the circumradius of the triangle (the radius of the circle passing through all three vertices).^[18] This law can be derived by dropping an altitude from one vertex to the opposite side, forming two right triangles, and applying the definition of sine in each; the common ratio emerges from the shared altitude length.^[18] The inclusion of $2R links the law directly to the circumcircle, providing a bridge to properties of the triangle's circumscribed circle.^[18] The law of cosines provides another fundamental relation for any triangle:

c^2 = a^2 + b^2 - 2ab \cos C,

with cyclic permutations for the other sides.^[1] This formula generalizes the Pythagorean theorem, recovering c^2 = a^2 + b^2 when C = 90^\circ since \cos 90^\circ = 0.^[19] Its derivation starts from the Pythagorean theorem applied to the two right triangles formed by dropping an altitude to side c, then using the cosine definition in one of those right triangles to account for the angle C.^[19] Together, the laws of sines and cosines enable the solution of triangles given partial information about sides and angles.

Associated Points, Lines, and Circles

In a triangle, several key points arise from the intersections of specific lines connecting vertices to sides or midpoints. The centroid is the intersection point of the three medians, where each median is divided in a 2:1 ratio, with the longer segment closer to the vertex.^[1] The orthocenter is the point where the three altitudes intersect, each altitude being the perpendicular from a vertex to the line containing the opposite side.^[1] The incenter forms at the intersection of the angle bisectors, which divide each vertex angle into two equal parts.^[1] The circumcenter is defined as the intersection of the perpendicular bisectors of the sides, equidistant from all three vertices.^[1] The lines associated with these points include the medians, which connect each vertex to the midpoint of the opposite side.^[1] The altitudes are the perpendicular lines from each vertex to the opposite side (or its extension in obtuse triangles).^[1] Angle bisectors emanate from each vertex, splitting the angle into two equal measures and meeting at the incenter.^[1] Perpendicular bisectors pass through the midpoint of each side and are perpendicular to it, converging at the circumcenter.^[1] Two prominent circles are derived from the triangle's elements: the incircle and the circumcircle. The incircle is tangent to all three sides, with its center at the incenter; its radius r is given by

r = \frac{A}{s},

where A is the area of the triangle and s is the semiperimeter.^[20] The circumcircle passes through all three vertices, centered at the circumcenter; its radius R, known as the circumradius, satisfies

\frac{a}{\sin A} = 2R,

or equivalently,

R = \frac{a}{2 \sin A},

where a is the side opposite angle A.^[21] In many triangles, the orthocenter, centroid, and circumcenter are collinear along the Euler line, with the centroid dividing the segment from orthocenter to circumcenter in a 2:1 ratio (longer part toward the orthocenter).^[22] This configuration holds for all non-equilateral triangles, providing a fundamental relation among these points.^[22]

Measurement and Relations

Area Formulas

The area of a triangle, denoted as K, represents the space enclosed by its three sides and can be calculated using several formulas based on the available measurements, such as side lengths, angles, or coordinates of vertices. These formulas derive from fundamental geometric principles and trigonometric identities, enabling computation in diverse contexts like surveying and computational geometry. One fundamental formula expresses the area in terms of two sides and the included angle. For a triangle with sides a and b enclosing angle C, the area is

K = \frac{1}{2} a b \sin C.

This formula arises from the base-height relation, where the height relative to base a is b \sin C, and it applies to any triangle via the law of sines for angle evaluation.^[23] For right-angled triangles, where the right angle is between legs of lengths a and b, the formula simplifies to

K = \frac{1}{2} a b,

as \sin 90^\circ = 1. This is a direct consequence of the general trigonometric formula and serves as a foundational case in Euclidean geometry.^[23] When all three side lengths a, b, and c are known, Heron's formula provides the area without requiring angles or heights:

K = \sqrt{s(s - a)(s - b)(s - c)},

where s = (a + b + c)/2 is the semiperimeter. Attributed to Heron of Alexandria, this formula appears in Proposition I.8 of his work Metrica (circa 60 CE) and is derived by inscribing a circle in the triangle to relate the area to the tangents from vertices to the points of tangency.^[24] In coordinate geometry, the area of a triangle with vertices at (x_1, y_1), (x_2, y_2), and (x_3, y_3) is given by the shoelace formula:

K = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|.

This determinant-based expression, also known as Gauss's area formula or the surveyor's formula, originates from Albrecht Ludwig Friedrich Meister's 1769 treatise Generalia de genesi figurarum planarum et inde pendentibus earum affectionibus and extends to polygons by summing trapezoidal areas.^[25] The area also relates to the triangle's incircle and circumcircle. Specifically,

K = r s,

where r is the inradius and s the semiperimeter; this follows from dissecting the triangle into three smaller triangles from the incenter, each with height r and bases summing to the perimeter.^[24] Likewise,

K = \frac{a b c}{4 R},

where R is the circumradius, derived from the extended law of sines (a = 2 R \sin A, etc.) substituted into the trigonometric area formula, yielding K = 2 R^2 (\sin A \sin B \sin C), which rearranges to the given form.^[26] These relations highlight connections between area and associated circles, useful in advanced geometric analysis.

Similarity and Congruence

In Euclidean geometry, two triangles are congruent if one can be superimposed onto the other through a sequence of rigid motions—translations, rotations, and reflections—such that corresponding vertices coincide, resulting in all corresponding sides and angles being equal in measure.^[27] The standard criteria for establishing triangle congruence without constructing the full superposition are the Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Hypotenuse-Leg (HL) postulates, applicable specifically to right triangles. Under SSS, if all three pairs of corresponding sides are equal, the triangles are congruent; SAS requires two sides and the included angle to be equal; ASA demands two angles and the included side; AAS specifies two angles and a non-included side; and HL states that for right triangles, equal hypotenuses and one corresponding leg suffice.^[28] These criteria ensure that once proven, all remaining corresponding parts—sides and angles—are equal by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem.^[29] A foundational proof for the SAS congruence criterion relies on rigid motions: given two triangles with equal corresponding sides AB = DE and AC = DF, and included angles \angle BAC = \angle EDF, one can translate and rotate the first triangle so that side AB coincides with DE, aligning the angles; the equal adjacent sides AC and DF then force vertex C to coincide with F under reflection if necessary, mapping the entire triangle rigidly.^[30] This approach, rooted in Euclid's Elements (Book I, Proposition 4), extends to the other criteria through similar transformations or auxiliary constructions.^[27] Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional, meaning there exists a positive scale factor k such that the ratios of all pairs of corresponding sides equal k.^[31] The criteria for similarity mirror congruence but incorporate proportionality: Angle-Angle (AA) requires two pairs of equal corresponding angles (the third follows from the angle sum property); Side-Side-Side (SSS) demands all three side ratios equal; and Side-Angle-Side (SAS) specifies two proportional sides with an included equal angle.^[32] These ensure the triangles have the same shape but possibly different sizes, with Euclid establishing AA and SAS~ in Elements Book VI, Propositions 4 and 5.^[31] For similar triangles, linear dimensions scale by k, while areas scale by k^2, as area is a two-dimensional measure derived from side lengths squared in formulas like Heron's.^[33] For instance, if \triangle ABC \sim \triangle DEF with k = 2, the perimeter of \triangle DEF doubles, but its area quadruples relative to \triangle ABC.^[34] These concepts apply in problem-solving by enabling the setup of proportional equations to determine unknown sides or angles; for example, in \triangle ABC \sim \triangle DEF with known \angle A = \angle D and side AB/DE = 3/4, one can compute missing lengths like BC/EF = 3/4 via cross-multiplication.^[35] Such proportions arise in real-world scenarios like shadow measurements for heights or map scaling, where similarity justifies indirect calculations without direct measurement.^[36]

Side Length Constraints

For three positive real numbers a, b, and c to serve as the side lengths of a non-degenerate triangle, they must satisfy the triangle inequality theorem, which states that the sum of any two sides must be strictly greater than the third side: a + b > c, a + c > b, and b + c > a.^[37] This condition ensures that the points forming the vertices are not collinear and enclose a positive area.^[37] One intuitive proof of the theorem proceeds by contradiction: suppose a + b \leq c; then, attempting to position the sides would force the endpoints of a and b to lie on or beyond the line segment of length c, resulting in collinear points and failing to form a closed triangular shape.^[37] An alternative geometric proof involves "unfolding" the triangle: consider traversing from one vertex to another via the third; the direct straight-line path (the third side) is the shortest distance, so the path along the other two sides must exceed it.^[37] For instance, in triangle ABC with sides a = BC, b = AC, c = AB, extending AB beyond B and reflecting point C shows that a + b forms a longer path than c.^[38] The strict inequality guarantees a positive area; when equality holds (e.g., a + b = c), the configuration degenerates into a straight line segment with zero area, known as a degenerate triangle.^[39] These constraints are essential for criteria like side-side-side (SSS) congruence, which assumes valid side lengths forming a triangle.^[37] Additionally, the side lengths determine the relative sizes of the angles: the largest side is opposite the largest angle, and the smallest side is opposite the smallest angle, as established in Euclidean geometry.^[40] For example, if c > a and c > b, then the angle opposite c exceeds the other two angles.^[41]

Structural Properties

Rigidity and Stability

In the Euclidean plane, a triangle is the simplest rigid polygon, meaning it cannot be continuously deformed while preserving the lengths of its sides, in contrast to quadrilaterals, which can undergo shearing motions without altering side lengths. This inherent rigidity arises because the three side lengths uniquely determine the shape up to congruence, as established by the side-side-side (SSS) congruence theorem. Any attempt to flex a triangular framework of rigid bars connected by pin joints would violate at least one distance constraint, preventing non-trivial deformations.^[42] This property can be formalized through the concept of degrees of freedom in framework theory. Each vertex in a planar structure contributes two degrees of freedom (its x and y coordinates), yielding 2n degrees for n vertices; however, three of these account for trivial rigid-body motions (two translations and one rotation), leaving 2n - 3 internal degrees of freedom that must be constrained for rigidity. For a triangle with n = 3 vertices and three bars, the constraints exactly match the 2(3) - 3 = 3 internal degrees, rendering it minimally rigid—no fewer bars suffice, and additional bars would introduce redundancy.^[42]^[43] In structural engineering, the rigidity of triangles underpins the stability of trusses and frameworks, where they serve as basic building blocks to distribute loads efficiently and resist deformation. Simple trusses begin with a single triangular unit and expand by adding panels that maintain triangular configurations, ensuring internal stability as each joint connects precisely three members. James Clerk Maxwell's 1864 rule for plane truss rigidity states that a structure with j joints is minimally rigid if it has exactly 2j - 3 bars, a condition satisfied by fully triangulated designs that eliminate mechanisms while avoiding overconstraint.^[44]^[45] For non-planar implications, triangles on curved surfaces like a sphere exhibit spherical excess—the positive difference between the sum of interior angles and π radians—which quantifies the enclosed area and alters stability relative to planar cases, with larger excesses corresponding to greater rigidity in geodesic frameworks. Complex planar structures can achieve overall rigidity by decomposition into such triangular elements.^[46]

Triangulation and Decomposition

Triangulation refers to the process of partitioning a simple polygon into a set of non-overlapping triangles that cover the entire interior of the polygon, achieved by adding non-intersecting diagonals between its vertices.^[47] This decomposition preserves the polygon's boundary and interior without introducing new vertices.^[47] A fundamental result in computational geometry states that every simple polygon with n vertices admits a triangulation, and any such triangulation consists of exactly n-2 triangles connected by n-3 non-intersecting diagonals.^[47] This theorem guarantees that simple polygons can always be decomposed into triangles without adding Steiner points or altering the original vertex set, enabling efficient geometric processing.^[47] The proof typically proceeds by induction, starting from the base case of a triangle (n=3) and showing that any larger polygon has an "ear"—a diagonal that splits it into a smaller polygon and a triangle—allowing recursive triangulation.^[47] Several algorithms exist for computing polygon triangulations, each balancing computational efficiency, implementation simplicity, and output quality. The ear-clipping method, based on the two-ears theorem asserting that every simple polygon with more than three vertices has at least two ears (convex vertices where the diagonal does not intersect the boundary), iteratively identifies and removes such ears to form triangles.^[48] This approach runs in O(n^2) time in the worst case but is straightforward to implement and suitable for non-convex polygons.^[48] In contrast, the constrained Delaunay triangulation extends the Delaunay criterion—where no point lies inside the circumcircle of any triangle—to respect polygon edges, producing a triangulation that maximizes the minimum angle among all possible triangulations of the vertex set. This property avoids skinny triangles, which can degrade numerical stability in simulations, and the algorithm can be computed in O(n \log n) time using incremental or sweep-line techniques. Polygon triangulation finds widespread applications across computational fields. In computer graphics, it facilitates mesh generation for rendering complex surfaces by breaking polygons into triangles compatible with GPU pipelines.^[49] Geographic information systems (GIS) employ triangulated irregular networks (TINs) derived from polygon decompositions to model terrain elevation and perform spatial queries efficiently.^[50] In finite element methods for numerical simulations, triangulations decompose domains into elements for approximating solutions to partial differential equations, with quality metrics like minimum angle ensuring convergence and accuracy.^[49] These decompositions also leverage the inherent rigidity of triangles to enhance the structural stability of the overall partitioned shape.^[47]

Spatial Positioning

Locating Points Relative to a Triangle

Determining the position of a point relative to a triangle in the plane is a fundamental task in geometry, enabling classifications such as interior, boundary, or exterior placement. These methods are essential for algorithms that process spatial data, such as rendering graphics or analyzing shapes. One key approach involves coordinate systems tailored to the triangle's vertices, while others rely on intersection or enclosure tests for efficient computation.^[51] Barycentric coordinates represent a point P within the plane of a triangle with vertices A, B, and C as a convex combination P = \alpha A + \beta B + \gamma C, where \alpha + \beta + \gamma = 1 and \alpha, \beta, \gamma \geq 0.^[51] The coefficients \alpha, \beta, \gamma indicate the relative areas of the sub-triangles formed by P and the opposite vertices, normalized by the total area of ABC.^[52] A point lies inside the triangle if all coordinates are positive (\alpha > 0, \beta > 0, \gamma > 0); on a boundary if one is zero and the others positive; and outside if at least one is negative.^[51] These coordinates can be computed by solving a linear system or using area ratios, providing an affine-invariant representation useful for interpolation and transformation.^[52] For containment testing without coordinates, ray casting shoots a ray from the point in a fixed direction (e.g., horizontal) and counts intersections with the triangle's edges; an odd count indicates the point is inside, while even indicates outside, assuming no ray-edge alignments.^[53] The winding number method, suitable for oriented boundaries, computes the total signed angle subtended by the triangle's edges around the point; a winding number of 1 (for counterclockwise orientation) confirms interior placement.^[54] These techniques extend the point-in-polygon problem to triangles, offering robustness for convex shapes like triangles with O(1) time complexity.^[55] An alternative brief check uses sub-triangle areas: if the sum of areas of triangles formed by the point and each edge equals the original triangle's area, the point is inside.^[54] The distance from a point to a triangle is zero if the point lies inside or on the boundary; otherwise, it is the minimum distance to the three edges or vertices.^[56] For points outside, project the point onto each edge (line segment) and select the closest projection, or to a vertex if the projection falls outside the segment.^[56] The distance to a line segment involves clamping the projection to the endpoints and computing the Euclidean distance, ensuring the nearest point on the boundary is found in constant time.^[56] In computational geometry, these positioning methods underpin point-in-triangle tests for mesh processing, collision detection, and finite element analysis, where rapid classification optimizes simulations of physical systems.^[52] Barycentric coordinates, in particular, facilitate smooth interpolation across triangular elements in graphics and numerical methods.^[51]

Triangles in Higher Dimensions

In higher-dimensional Euclidean spaces, the concept of a triangle generalizes to the 2-simplex, which is the convex hull of three affinely independent points in \mathbb{R}^n for n \geq 2. These points define the affine span of the simplex, a 2-dimensional affine subspace embedded within the higher-dimensional space, preserving the triangle's topological and geometric properties such as convexity and boundary structure.^[57]^[58] In three-dimensional Euclidean space, triangles formed by three points remain inherently planar, as any three non-collinear points uniquely determine a plane, making non-planar or "skewed" triangles invalid by definition.^[59] This planarity ensures that the triangle lies flat within its defining plane, regardless of the surrounding 3D environment. On curved surfaces like spheres, however, spherical triangles emerge, where great-circle arcs connect vertices on the sphere's surface, resulting in an angle sum exceeding $180^\circ. According to Girard's theorem, the spherical excess E = A + B + C - \pi (in radians) equals the triangle's area divided by the sphere's radius squared, E = \frac{\text{area}}{R^2}, linking angular properties directly to surface measure.^[60] In non-Euclidean geometries, hyperbolic triangles arise in spaces where the parallel postulate fails, allowing multiple parallels through a point not on a given line, which leads to angle sums less than $180^\circ. This angle defect reflects the geometry's negative curvature and distinguishes hyperbolic triangles from their Euclidean counterparts, with properties such as asymptotic behavior of sides emerging from the absence of the parallel postulate.^[61] In modern applications, particularly machine learning, simplicial complexes extend triangular structures to higher dimensions for topological data analysis, enabling triangulation of datasets to capture multi-scale features and persistent homology. These complexes represent data as higher-dimensional analogs of graphs, facilitating tasks like shape inference and anomaly detection in complex datasets.^[62]^[63]

Extensions and Variants

Inscribed and Circumscribed Figures

In a triangle, an inscribed figure is one that lies entirely within the triangle and touches its boundary, typically at the sides or vertices. The most prominent example is the incircle, which is tangent to all three sides of the triangle. Every triangle possesses a unique incircle, as the point of tangency on each side is determined by the semiperimeter minus the opposite side length, ensuring a single solution for the incenter where the angle bisectors intersect.^[64] The radius r of the incircle can be expressed in relation to the angles and semiperimeter as r = (s - a) \tan(A/2), where s is the semiperimeter, a is the side opposite angle A, and this formula arises from the geometry of the tangential segments from the incenter to the vertices.^[65] Beyond circles, polygons can also be inscribed in a triangle by having their vertices lie on the triangle's sides. A key example is the midpoint triangle, formed by connecting the midpoints of the original triangle's sides. This medial triangle is similar to the original triangle, with side lengths half as long and an area one-fourth that of the original, and its sides are parallel to the original triangle's sides due to the midpoint theorem.^[66] The midpoint triangle divides the original into four congruent smaller triangles of equal area, providing a useful decomposition for geometric analysis.^[67] Certain theorems highlight properties of points and distances related to inscribed figures. Viviani's theorem applies specifically to equilateral triangles, stating that the sum of the perpendicular distances from any interior point to the three sides equals the altitude of the triangle.^[68] This constant sum reflects the uniform tangential nature of the incircle in such symmetric cases and extends conceptually to the equidistance property from the incenter. For circumscribed figures, which enclose the triangle such that the triangle touches their boundary from inside, the circumcircle is the unique circle passing through all three vertices of the triangle. Its center, the circumcenter, is the intersection of the perpendicular bisectors of the sides, and it exists for any non-degenerate triangle, serving as the smallest circle enclosing the vertices.^[69] While polygons circumscribing a triangle—such as quadrilaterals with the triangle's vertices on their sides—can vary, tangential quadrilaterals relate indirectly through shared incircles; for instance, a triangle's incircle can be the incircle of a tangential quadrilateral formed by extending the sides, though this construction is not unique and depends on the choice of tangent points.^[70]

Special and Miscellaneous Triangles

A golden triangle is an isosceles triangle in which the two longer sides are in the golden ratio \phi = (1 + \sqrt{5})/2 \approx 1.618 to the base, resulting in angles of $72^\circ, $72^\circ, and $36^\circ.^[71] This configuration arises naturally in the construction of regular pentagons and exhibits self-similar properties when bisected, as the resulting triangles are similar to the original.^[72] Heronian triangles are those with rational side lengths and rational area, named after Heron of Alexandria due to their connection with Heron's formula for area computation.^[73] When scaled appropriately, they possess integer side lengths and integer area, enabling exact constructions on lattice points; examples include the (5,5,6) triangle with area 12 and the (3,4,5) triangle with area 6. These triangles are significant in number theory for parametrizing integer solutions to the triangle inequality with rational areas. A perfect triangle is defined as a Heronian triangle that additionally has rational medians and altitudes.^[74] Since rational sides and area imply rational altitudes via h_a = 2K / a (where K is the area), the key additional condition is rational medians; it remains an open conjecture that no such triangle exists, as supported by exhaustive searches and partial proofs showing contradictions in certain cases.^[75] Every non-degenerate triangle is cyclic, meaning its vertices lie on a unique circle known as the circumcircle, with radius given by R = abc / (4K). Pseudotriangles, in the context of computational geometry, are simple polygons featuring exactly three convex vertices connected by three chains of reflex (concave) edges, generalizing traditional triangles to handle non-convex boundaries while preserving certain triangulation properties.^[76] The three sides consist of chains of straight line segments that form concave boundaries, making them useful for partitioning point sets without full convexity. The Sierpinski triangle is a fractal and self-similar set formed by starting with an equilateral triangle and iteratively removing the open central subtriangle at each stage, connecting the midpoints of the sides. This process yields a figure with uncountably many points but area approaching zero, and its Hausdorff dimension is \log_2 3 \approx 1.58496, reflecting its space-filling behavior between one and two dimensions.

\begin{aligned} &\dim_H(S) = \frac{\log 3}{\log 2}, \end{aligned}

where S is the Sierpinski triangle and \dim_H denotes the Hausdorff dimension.^[77] Napoleon's theorem, discovered in 1825, states that erecting equilateral triangles outwardly (or inwardly) on the sides of any triangle results in the centroids of those equilateral triangles forming another equilateral triangle.^[78] This holds regardless of the original triangle's shape, with the resulting Napoleon triangle having side lengths related by complex number rotations or vector sums.^[79] Morley's trisector theorem, proved in 1899, reveals that the intersection points of the adjacent trisectors from each angle of any triangle form an equilateral triangle, known as the Morley triangle.^[80] The side lengths of this inner equilateral triangle scale with the original angles, and in specific cases like a $30^\circ-$60^\circ-$90^\circ triangle, the trisector intersections align with notable angular relations, though the core result is equilateral symmetry.^[81] Spherical triangles, formed by the intersections of three great circles on a sphere, generalize planar triangles to curved surfaces and include variants bounded by lunes (spherical digons formed by two great circles), which create lune-delimited regions resembling triangles in non-Euclidean geometry.

Applications in Other Fields

In physics and engineering, triangles facilitate the resolution of vectors into components, enabling the analysis of forces and motions. For instance, vector resolution decomposes a force into perpendicular components using right triangles, where the magnitudes are determined via trigonometric functions such as sine and cosine.^[82] Similarly, the triangle of forces represents three coplanar forces in equilibrium at a point, with each side of the triangle proportional to a force's magnitude and direction, a method applied in statics to ensure structural balance.^[83] Triangulation has been a cornerstone of surveying for large-scale mapping since the 16th century, involving the measurement of angles in a network of triangles to determine distances and positions across terrain.^[84] In modern navigation, the Global Positioning System (GPS) employs trilateration, a spherical analog of triangulation, where distances from multiple satellites intersect to locate a receiver, forming implicit spherical triangles on Earth's surface.^[85] In computer science, triangle meshes form the basis of 3D rendering in graphics, approximating complex surfaces with interconnected triangles for efficient rasterization and shading in real-time applications like video games and simulations.^[86] Delaunay triangulation, the dual of Voronoi diagrams, generates optimal meshes by maximizing minimum angles, with applications in terrain modeling, finite element analysis, and computational geometry for partitioning spaces.^[87] Triangles underpin perspective drawing in art, where converging lines to vanishing points create depth, often constructed using triangular guides to align objects in linear perspective systems developed during the Renaissance.^[88] In architecture, Pythagorean tilings—repetitive patterns of squares on right-triangle sides—illustrate the Pythagorean theorem visually and appear in floor mosaics and decorative elements, as seen in historical sites like those in Huesca, Spain.^[89] Recent advancements in artificial intelligence leverage triangular structures in transformer models; for example, triangle attention in AlphaFold2 encodes geometric constraints like the triangle inequality for pairwise residue interactions, enabling highly accurate protein structure predictions. In quantum computing, post-2020 research explores simplex states, where quantum walks on simplex graphs enable multiple perfect state transfers among vertices, advancing fault-tolerant quantum information processing.^[90] In biology, triangular lattices approximate molecular structures, such as in DNA origami designs where tensegrity triangles self-assemble into 2D or 3D crystals, facilitating nanotechnology applications like drug delivery scaffolds.^[91]

References

[1]
Triangle -- from Wolfram MathWorld
A triangle is a 3-sided polygon with three sides and three angles. It can be acute, obtuse, right, equilateral, isosceles, or scalene.
[2]
Triangle definition and properties - Math Open Reference
A closed figure consisting of three line segments linked end-to-end. A 3-sided polygon. Try this Drag the orange dots on each vertex to reshape the triangle.
[3]
Euclid's Elements, Book I, Guide - cs.clarku.edu
14) a triangle is a two-dimensional figure bounded by three straight lines as sides. There are actually a lot of subtleties in these definitions, but in a ...
[4]
Laws of Cosines & Sines - cs.clarku.edu
As usual, we'll use a standard notation for the angles and sides of a triangle. That means the side a is opposite the angle A, the side b is opposite the ...
[5]
Triangle - Etymology, Origin & Meaning
Triangle, from Latin triangulum meaning "three-cornered," combines tri- (three) + angulus (angle), denoting a three-sided rectilineal geometric figure.
[6]
Babylonians used Pythagorean theorem 1,000 years before it was ...
Sep 22, 2022 · A 3,700-year-old clay tablet has revealed that the ancient Babylonians understood the Pythagorean theorem more than 1,000 years before the ...
[7]
Triangles - Equilateral, Isosceles and Scalene - Math is Fun
Equilateral Triangle. Three equal sides. Three equal angles, always 60° ; Isosceles Triangle. Two equal sides. Two equal angles ; Scalene Triangle. No equal sides
[8]
Classifying Triangles by Sides or Angles - Geometry - CliffsNotes
Triangles can be classified either according to their sides or according to their angles. All of each may be of different or the same sizes.
[9]
Types of triangles review (article) - Khan Academy
Classifying triangles by their angles ; Acute triangles. An acute · triangle has 3 angles that ‍ each measure less than ; Right triangles. A right · triangle has 1 ...
[10]
Triangle Classification | CK-12 Foundation
Every triangle has three vertices (the points where the segments meet), three sides (the segments), and three interior angles (formed at each vertex).
[11]
Types of Triangles - Definition, Properties, Examples - SplashLearn
There are 3 types of triangles based on the measure of their sides. We classify the triangles by their sides as equilateral, isosceles, and scalene.
[12]
Types Of Triangles - BYJU'S
There are basically six different types of triangles with respect to the length and measure of the lines and angles of a triangle, respectively.
[13]
Degenerate - Mathwords
A degenerate triangle is the "triangle" formed by three collinear points. It ... Degenerate examples can be used to test the general applicability of ...
[14]
The Triangle Inequality and Degenerate Triangles - Expii
The triangle is degenerate if →a∥→b, and indeed, if →a and →b point in the same direction, equality holds. So how do we prove the triangle inequality? Suppose ...Missing: definition | Show results with:definition<|control11|><|separator|>
[15]
Angles in Triangle Add to 180: history and a collection of proofs
In any triangle, the three interior angles are equal to two right angles, and the exterior angle is equal to the sum of the two interior and opposite angles.
[16]
1.5: Triangles - Mathematics LibreTexts
Sep 4, 2021 · Theorem An exterior angle is equal to the sum of the two remote interior angles, In Figure 1 . 5 . 4 , ∠ x = ∠ A + ∠ B .
[17]
Sine Rule (Law of Sines) | Brilliant Math & Science Wiki
The law of sines is a relationship linking the sides of a triangle with the sine of their corresponding angles. Given the following triangle ...
[18]
8.2: Non-right Triangles - Law of Cosines - Mathematics LibreTexts
Jan 2, 2021 · Understanding how the Law of Cosines is derived will be helpful in using the formulas. The derivation begins with the Generalized Pythagorean ...
[19]
Inradius -- from Wolfram MathWorld
Rr_d=rho^2. (14). See also. Carnot's Theorem, Circumradius, Euler Triangle Formula, Japanese Theorem, Midradius. Explore with Wolfram|Alpha. WolframAlpha. More ...Missing: perimeter | Show results with:perimeter
[20]
[PDF] 1 Triangles
Extended Law of Sines a/ sin A = 2R. 2. [ABC] = abc/4R. 3. (Geometry ... Orthic Triangle The feet of the altitudes of triangle ABC determine a triangle, called ...
[21]
Euler Line -- from Wolfram MathWorld
AP^2+BP^2+CP^2, = 3R^2+((k-4)kOH^2)/(. (13). = 3R^2+((k-4)OP^2)/k. (14). where R is the circumradius, O is the circumcenter, and H is the orthocenter of the ...<|separator|>
[22]
Area of a triangle - cs.clarku.edu
There are several ways to compute the area of a triangle. For instance, there's the basic formula that the area of a triangle is half the base times the height.Missing: derivation | Show results with:derivation
[23]
[PDF] Heron's Formula for Triangular Area - Mathematics
Heron's Formula can be used as a proof of the Pythagorean Theorem. Pythagorean Theorem from. Heron's Formula. Suppose we have a right triangle with hypotenuse.
[24]
[PDF] Who invented the Shoelace Formula? - Theorem of the Day
Basic fact: the area of the triangle is the cross product of the direction vectors of any two sides in anticlockwise order. Area = 1. 2. (−v0 + v1) × (−v1 + v2) ...
[25]
[PDF] ORMC AMC 10/12 Group (More) Circles
Jan 15, 2023 · If we recall the (extended) law of sines: c sin(C). = b sin(B). = a sin(A). = abc. 2K. =⇒ R = abc. 4K . 2. Page 3. 2.1 Examples. 1. (2004 AMC ...
[26]
Euclid's Elements, Book I, Proposition 4 - Clark University
This is the first of the congruence propositions for triangles. Euclid did not explicitly use the concept of congruence, although it would have simplified his ...
[27]
Triangle congruence postulates/criteria (video) - Khan Academy
May 27, 2020 · Sal introduces and justifies the SSS, SAS, ASA and AAS postulates for congruent triangles. He also shows that AAA is only good for similarity.Missing: sources | Show results with:sources
[28]
Methods of Proving Triangle Congruent - MathBitsNotebook(Geo)
Once triangles are proven congruent, the corresponding leftover "parts" that were not used in SSS, SAS, ASA, AAS and HL, are also congruent. CPCTC Corresponding ...Missing: sources | Show results with:sources
[29]
Proving the SAS triangle congruence criterion using transformations
Jan 27, 2021 · If those have the same lengths or measures, then we can deduce that these two triangles must be congruent by the rigid motion definition of congruency. Or ...
[30]
Euclid's Elements, Book VI, Definition 1 - Clark University
Propositions VI.4 and VI.5 give two criteria for two triangles to be similar. Proposition VI.4 says that condition 1 implies similarity, while VI.5 ...
[31]
Similar Triangles - Formulas, Properties, Theorems, Proofs - Cuemath
Similar Triangles Theorems · AA (or AAA) or Angle-Angle Similarity Criterion · SAS or Side-Angle-Side Similarity Criterion · SSS or Side-Side-Side Similarity ...
[32]
Area of Similar Triangles Theorem - BYJU'S
Theorem: If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.Missing: scale factor
[33]
Similar Triangles: Perimeters and Areas - Geometry - CliffsNotes
Theorem 61: If two similar triangles have a scale factor of a : b, then the ratio of their areas is a2 : b2. Example 2: In Figure 4, Δ PQR∼ Δ STU. Find the area ...Missing: source | Show results with:source
[34]
Solving similar triangles (video) - Khan Academy
Aug 9, 2016 · If you start with the ratio of the larger triangle to the ... Can't you just do 5+3=8, then divide 8 by three, and multiply the scale factor by four?
[35]
Triangle Similarity Theorems (23 Examples for Mastery!)
Jan 21, 2020 · The three triangle similarity theorems are AA, SAS, and SSS. AA uses two angles, SAS uses two sides and an included angle, and SSS uses all ...
[36]
Triangle Inequality | Brilliant Math & Science Wiki
The triangle inequality states that the sum of the lengths of any two sides of a triangle is greater than the length of the remaining side.
[37]
Triangle Inequality Theorem, Proof & Applications - GeeksforGeeks
Jul 23, 2025 · The Triangle Inequality Theorem is proven by extending a side of a triangle and applying angle and side properties to establish the inequality ...
[38]
Triangle Inequality - AoPS Wiki
### Summary of Triangle Inequality from AoPS Wiki
[39]
Euclid's Elements, Book I, Proposition 18 - Clark University
In any triangle, the angle opposite the greater side is greater. For example, if AC > AB, then angle ABC > angle BCA.
[40]
Inequalities in Triangles
An exterior angle is greater than any interior angle. Greater sides are opposite greater angles. Greater sides imply greater opposite angles.
[41]
[PDF] the geometry of frameworks: rigidity, mechanisms and cad - WPI
An easy measure of the deformability of a framework is its degree of freedom. A point in the plane has two degrees of freedom, say its x and y coordinate, while ...
[42]
[PDF] The basics of rigidity - Cornell Mathematics
The rigidity matrix R(p) = df is the differential of f. Basic general theorem: If a (bar) framework is infinitesimally rigid in Ed, then it is rigid in Ed.
[43]
[PDF] Analysis of Statically Determinate Trusses
□ Simple Truss. □ To prevent collapse, the framework of a truss must be rigid. □ The simplest framework that is rigid or stable is a triangle. Page 10 ...
[44]
L. On the calculation of the equilibrium and stiffness of frames
On the calculation of the equilibrium and stiffness of frames. J. Clerk Maxwell FRS. King's College, London. Pages 294-299 | Published online: 26 May 2009.
[45]
Spherical Excess -- from Wolfram MathWorld
Spherical excess is the difference between the sum of angles (A+B+C) of a spherical triangle and pi radians (180 degrees), calculated as E=A+B+C-pi.
[46]
[PDF] Polygon Triangulation (chapter 3) - Purdue Computer Science
Theorem 3.1 Every n-vertex polygon has a triangulation and every triangulation has n − 2 triangles. Proof. ▷ A triangle (n = 3) is trivial, so consider n > 3.<|control11|><|separator|>
[47]
[PDF] Triangulation by Ear Clipping - Geometric Tools
Nov 18, 2002 · This pseudosimple polygon is then triangulated using the ear clipping algorithm to produce triangles involving all Vj for 0 ≤ j ≤ 13. The ...
[48]
Computational Geometry - Some Easy
Apr 14, 2001 · Main fields of application include finite element methods and computer aided design. In formulating a triangulation problem, a choice arises ...
[49]
The Methods of Triangulation - Geospatial World
Sep 1, 2009 · Triangulation is a method for tessellation of domain. In fact triangulation is a common method for surface representation and for building a TIN.
[50]
[PDF] Barycentric Coordinates for Triangles
Barycentric coordinates provide a measuring system that positions a point relative to the edges of the triangle, with a very simple scheme for converting from ...
[51]
[PDF] Computational Geometry Lab: BARYCENTRIC COORDINATES IN ...
Aug 28, 2018 · The barycentric coordinate system gives us a kind of standard chart, which applies to all triangles, and allows us to determine the positions ...
[52]
[PDF] Search and Intersection
Point-Polygon/Polyhedron (2D/3D). » Winding Number. » Parity Test. Page 34. Point-Polygon/Polyhedron. Winding Number: Given a point 𝑝 and a curve 𝐶 in the ...
[53]
[PDF] Ray Casting II - DSpace@MIT
Winding number. • To solve problem with star pentagon. • Oriented edges ... Use signed areas for points outside the triangle c a b. P Ta. T. Page 40. MIT ...
[54]
[PDF] • • •
solve this problem: ray casting and winding number. 7.1 Ray Casting. In ray casting, we fire a ray from the given point P to the fixed polygon S. If the point ...
[55]
[PDF] Rapid and Accurate Computation of the Distance Function Using Grids
This motivates our next generalization of the closest point algorithm. Given a triangle (or line segment ت ) in 024 and a point t , the following algorithm.
[56]
2.1 General Definitions
A simplex is the convex hull of a set of (n + 1) affinely independent points in some Euclidean space of dimension n or higher.
[57]
[PDF] Lecture 3. Convex Sets
Polyhedra are convex sets. the 0-simplex is just a point; the 1-simplex is a segment; the 2-simplex is a triangle; the 3-simplex is a tetrahedron; ……
[58]
Three Dimensional Geometry - World Web Math: Vector Calculus - MIT
Definition Given any three non-colinear points A,B, and C, then the set of all points D such that {A,B,C,D} is bounded coplanar is called the plane through A, B ...
[59]
Girard's Theorem
area ( T ) = R 2 ( r + g + b − π ) . The sum of the angles minus π radians is called the angle excess. This last formula is called Girard's formula, and ...
[60]
Hyperbolic Geometry – Math Fun Facts
... sum of angles in a triangle is less than 180 degrees? Yes! For instance ... Unlike planar geometry, the parallel postulate does not hold in hyperbolic geometry.
[61]
An Introduction to Topological Data Analysis - Frontiers
Simplicial complexes can be seen as higher-dimensional generalizations of neighboring graphs that are classically built on the top of data in many standard data ...
[62]
[PDF] simplicial complex representation learning - arXiv
Feb 2, 2022 · Simplicial complex representation learning embeds a complex into a space, preserving complex-to-complex proximity, using a complex autoencoder ...
[63]
Incircle -- from Wolfram MathWorld
Using the incircle of a triangle as the inversion center, the sides of the triangle and its circumcircle are carried into four equal circles (Honsberger 1976, p ...
[64]
https://mathworld.wolfram.com/Incircle.html
[65]
Medial Triangle -- from Wolfram MathWorld
### Properties of the Midpoint (Medial) Triangle in a Triangle
[66]
Triangle Midsegment Theorem - Varsity Tutors
In a triangle, a midsegment connecting the midpoints of two sides is parallel to the third side and is half as long as that side: D E ― ∥ B C ― and ...
[67]
Viviani's Theorem: New in Wolfram Language 12
Viviani's theorem states that in an equilateral triangle, the sum of the distances from any interior point to the three sides is equal to the height of the ...
[68]
Circumcircle -- from Wolfram MathWorld
The circumcircle is a triangle's circumscribed circle, ie, the unique circle that passes through each of the triangle's three vertices.
[69]
Tangential Quadrilateral -- from Wolfram MathWorld
A tangential quadrilateral has an incircle, where a single circle is tangent to all four sides. A rhombus is a special case.
[70]
Golden Triangle -- from Wolfram MathWorld
The golden triangle, sometimes also called the sublime triangle, is an isosceles triangle such that the ratio of the hypotenuse a to base b is equal to the ...
[71]
Golden Ratio in Geometry
In passing, the isosceles 72^{\circ}-36^{\circ}-72^{\circ} and 36^{\circ}-108^{\circ}-36^{\circ} triangles are known as the golden triangles because, as you ...<|control11|><|separator|>
[72]
Heronian Triangle -- from Wolfram MathWorld
Heronian Triangle ; a, = n(m^2+k^2) ; b, = m(n^2+k^2) ; c, = (m+n)(mn-k^2) ; s, = mn(m+n) ; Delta, = kmn(m+n)(mn-k^2).
[73]
[PDF] Cool-Math-Essay-Summer-2021-Heronian ... - James Tanton
Definition: A triangle with side lengths and area each a rational value is called. Heronian. Any Heronian triangle in this sense can be scaled to one with ...
[74]
Rational triangles
I guess it's the simplest rational triangle formed from two dissimilar integer-sided right triangles. There's a section in Maurice Kraitchik's "Mathematical ...
[75]
Heron Triangles and the Hunt for Unicorns
Mar 21, 2024 · ... integer sides, three integer medians, and integer area. Does a perfect triangle exist, or equivalently, is there a Heron triangle with three ...
[76]
[PDF] Pseudo-Triangulations — a Survey - Freie Universität Berlin
Abstract. A pseudo-triangle is a simple polygon with exactly three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region.
[77]
How to compute the dimension of a fractal | plus.maths.org
Jul 18, 2024 · This way of computing dimension works well for fractals which are exactly self-similar, such as the Sierpiński triangle or the Koch curve.
[78]
Napoleon's Theorem -- from Wolfram MathWorld
If equilateral triangles DeltaABE_(AB), DeltaBCE_(BC), and DeltaACE_(AC) are erected externally on the sides of any triangle DeltaABC, then their centers ...
[79]
Napoleon's Theorem, Two Simple Proofs
Napoleon's Theorem states that if equilateral triangles are built on each side of a triangle, the centroids of these triangles form an equilateral triangle.
[80]
Morley's Theorem -- from Wolfram MathWorld
Morley's Theorem states that the intersection points of a triangle's adjacent angle trisectors form an equilateral triangle.
[81]
An Elementary Proof of Morley's Trisector Theorem
Morley's theorem states that if ABC be any triangle, and if those trisectors of the angles B and C adjacent to BC meet in L,.
[82]
Vector Resolution: Finding the Components of a Vector
Vector resolution is the process of graphically or trigonometrically determining the magnitude and direction of a vector's components.<|separator|>
[83]
Lesson Explainer: Triangle of Forces - Nagwa
When a rigid body is in equilibrium under the action of three coplanar forces meeting at a point, we can analyze the situation using a triangle of forces.
[84]
Triangulation, Traverse and Horizontal Networks | NGS INFO
Aug 7, 2020 · In 1871, a major effort, known as the Transcontinental Arc of Triangulation, was undertaken to conduct a very precise survey across the country.
[85]
Trilateration | GPS.gov
Trilateration is determining a position by knowing your distance from at least 3 known points. In GPS those known points are the satellites themselves. It is ...
[86]
3.6 Triangle Meshes
This triangle mesh contains over four million individual triangles. It was created from a real statue using a 3D scanner that uses structured light to ...
[87]
https://www.csun.edu/~ctoth/Handbook/chap27.pdf
[88]
Geometry in Art & Architecture Unit 5 - Dartmouth Mathematics
This unit covers polygons (triangles, squares, hexagons), tilings, and sacred geometry, focusing on the equilateral triangle, hexagon, and pentagram.
[89]
Pythagoras' Theorem in a tiling - Matematicas Visuales
The tiling uses squares representing the legs of a right triangle. The theorem is shown by the two tiles having the same area.
[90]
Quantum walks on simplexes and multiple perfect state transfer
Various state transfers can be achieved on these graphs, such as multiple state transfer among extreme points of a simplex, fractional revival on subsimplexes.
[91]
Triangular DNA Origami Tilings - ACS Publications
Dec 4, 2018 · DNA origami tilings provide methods for creating complex molecular patterns and shapes using flat DNA origami structures as building blocks.