Category: Theorems about triangles and circles

Harcourt's theorem
Harcourt's theorem is a formula in geometry for the area of a triangle, as a function of its side lengths and the perpendicular distances of its vertices from an arbitrary line tangent to its incircle
Conway circle theorem
In plane geometry, the Conway circle theorem states that when the sides meeting at each vertex of a triangle are extended by the length of the opposite side, the six endpoints of the three resulting l
Carnot's theorem (inradius, circumradius)
In Euclidean geometry, Carnot's theorem states that the sum of the signed distances from the circumcenter D to the sides of an arbitrary triangle ABC is where r is the inradius and R is the circumradi
Feuerbach point
In the geometry of triangles, the incircle and nine-point circle of a triangle are internally tangent to each other at the Feuerbach point of the triangle. The Feuerbach point is a triangle center, me
Van Schooten's theorem
Van Schooten's theorem, named after the Dutch mathematician Frans van Schooten, describes a property of equilateral triangles. It states: For an equilateral triangle with a point on its circumcircle t
Trillium theorem
In Euclidean geometry, the trillium theorem – (from Russian: лемма о трезубце, literally 'lemma about trident', Russian: теорема трилистника, literally 'theorem of trillium' or 'theorem of trefoil') i
Lester's theorem
In Euclidean plane geometry, Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle.The result is named after J
Reuschle's theorem
In elementary geometry, Reuschle's theorem describes a property of the cevians of a triangle intersecting in a common point and is named after the German mathematician Karl Gustav Reuschle (1812–1875)
Miquel's theorem
Miquel's theorem is a result in geometry, named after , concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides. It is one of sev
Pompeiu's theorem
Pompeiu's theorem is a result of plane geometry, discovered by the Romanian mathematician Dimitrie Pompeiu. The theorem is simple, but not classical. It states the following: Given an equilateral tria
Thales's theorem
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the insc
Thébault's theorem
Thébault's theorem is the name given variously to one of the geometry problems proposed by the French mathematician Victor Thébault, individually known as Thébault's problem I, II, and III.
Musselman's theorem
In Euclidean geometry, Musselman's theorem is a property of certain circles defined by an arbitrary triangle. Specifically, let be a triangle, and , , and its vertices. Let , , and be the vertices of
Equal incircles theorem
In geometry, the equal incircles theorem derives from a Japanese Sangaku, and pertains to the following construction: a series of rays are drawn from a given point to a given line such that the inscri
Euler's theorem in geometry
In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by or equivalentlywhere and denote the circumradius and inradius respectively (the
Kosnita's theorem
In Euclidean geometry, Kosnita's theorem is a property of certain circles associated with an arbitrary triangle. Let be an arbitrary triangle, its circumcenter and are the circumcenters of three trian
Japanese theorem for cyclic polygons
In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant. Conversely, if the sum of inradii is independent of the tria