Category: Theorems about triangles

Stewart's theorem
In geometry, Stewart's theorem yields a relation between the lengths of the sides and the length of a cevian in a triangle. Its name is in honour of the Scottish mathematician Matthew Stewart, who pub
Desargues's theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states: Two triangles are in perspective axially if and only if they are in perspective centrally. Denote the three vertices
Carnot's theorem (conics)
Carnot's theorem (named after Lazare Carnot) describes a relation between conic sections and triangles. In a triangle with points on the side , on the side and on the side those six points are located
Apollonius's theorem
In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that "the sum of the squares of any two sides of any triangle equals
Droz-Farny line theorem
In Euclidean geometry, the Droz-Farny line theorem is a property of two perpendicular lines through the orthocenter of an arbitrary triangle. Let be a triangle with vertices , , and , and let be its o
Sylvester's triangle problem
Sylvester's theorem or Sylvester's formula describes a particular interpretation of the sum of three pairwise distinct vectors of equal length in the context of triangle geometry. It is also referred
Law of cotangents
In trigonometry, the law of cotangents is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles. This is also known as the Cot Theorem. Just
Carnot's theorem (perpendiculars)
Carnot's theorem (named after Lazare Carnot) describes a necessary and sufficient condition for three lines that are perpendicular to the (extended) sides of a triangle having a common point of inters
Thomsen's theorem
Thomsen's theorem, named after Gerhard Thomsen, is a theorem in elementary geometry. It shows that a certain path constructed by line segments being parallel to the edges of a triangle always ends up
Bottema's theorem
Bottema's theorem is a theorem in plane geometry by the Dutch mathematician (Groningen, 1901–1992). The theorem can be stated as follows: in any given triangle , construct squares on any two adjacent
Menelaus's theorem
Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ABC, and a transversal line that crosses BC, AC, and AB at points D
Mollweide's formula
In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle. A variant in more geometrical style was first published by Isaac Newton in 1707 and then by in 1
Napoleon's theorem
In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centres of those equilater
Law of cosines
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using nota
Maxwell's theorem (geometry)
Maxwell's theorem is the following statement about triangles in the plane. For a given triangle and a point not on the sides of that triangle construct a second triangle , such that the side is parall
Heron's formula
In geometry, Heron's formula (or Hero's formula) gives the area A of a triangle in terms of the three side lengths a, b, c. If is the semiperimeter of the triangle, the area is, It is named after firs
Morley's trisector theorem
In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley trian
Angle bisector theorem
In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their re
Pappus's area theorem
Pappus's area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. The theorem, which can also be thought of as a generalizati
Saccheri–Legendre theorem
In absolute geometry, the Saccheri–Legendre theorem states that the sum of the angles in a triangle is at most 180°. Absolute geometry is the geometry obtained from assuming all the axioms that lead t
Jacobi's theorem (geometry)
In plane geometry, a Jacobi point is a point in the Euclidean plane determined by a triangle ABC and a triple of angles α, β, and γ. This information is sufficient to determine three points X, Y, and
Law of tangents
In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, a, b, and c are the leng
Midpoint theorem (triangle)
The midpoint theorem or midline theorem states that if you connect the midpoints of two sides of a triangle then the resulting line segment will be parallel to the third side and have half of its leng
Routh's theorem
In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle points
Ceva's theorem
Ceva's theorem is a theorem about triangles in plane geometry. Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O (not on one of the sides of ABC), to mee
Exterior angle theorem
The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angl
Law of sines
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, where a, b, a
Marden's theorem
In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with comp