Nagel pointIn geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concurrency of all three of the triangle's splitters. Given a triangle △ABC, let T_A, T_B, T_C be the extouch points in which the A-excircle meets line BC, the B-excircle meets line CA, and the C-excircle meets line AB, respectively. The lines AT_A, BT_B, CT_C concur in the Nagel point N of triangle △ABC.
Cleaver (geometry)In geometry, a cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. They are not to be confused with splitters, which also bisect the perimeter, but with an endpoint on one of the triangle's vertices instead of its sides. Each cleaver through the midpoint of one of the sides of a triangle is parallel to the angle bisectors at the opposite vertex of the triangle. The broken chord theorem of Archimedes provides another construction of the cleaver.
RadiusIn classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the Latin radius, meaning ray but also the spoke of a chariot wheel. The typical abbreviation and mathematical variable name for radius is R or r. By extension, the diameter D is defined as twice the radius: If an object does not have a center, the term may refer to its circumradius, the radius of its circumscribed circle or circumscribed sphere.
Area of a circleIn geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter pi represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159. One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons with an increasing number of sides.
Circular sectorA circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle, the radius of the circle, and is the arc length of the minor sector. The angle formed by connecting the endpoints of the arc to any point on the circumference that is not in the sector is equal to half the central angle.
Equilateral polygonIn geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon does not need to be a convex polygon: it could be concave or even self-intersecting. All regular polygons and edge-transitive polygons are equilateral.
CevianIn geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovanni Ceva, who proved a well-known theorem about cevians which also bears his name.
SemicircleIn mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180° (equivalently, pi radians, or a half-turn). It has only one line of symmetry (reflection symmetry). In non-technical usage, the term "semicircle" is sometimes used to refer to either a closed curve that also includes the diameter segment from one end of the arc to the other or to the half-disk, which is a two-dimensional geometric region that further includes all the interior points.
Curve of constant widthIn geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or an orbiform, the name given to these shapes by Leonhard Euler. Standard examples are the circle and the Reuleaux triangle. These curves can also be constructed using circular arcs centered at crossings of an arrangement of lines, as the involutes of certain curves, or by intersecting circles centered on a partial curve.
Heronian triangleIn geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths a, b, and c and area A are all positive integers. Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula which Heron demonstrated with the example triangle of sides 13, 14, 15 and area 84. Heron's formula implies that the Heronian triangles are exactly the positive integer solutions of the Diophantine equation that is, the side lengths and area of any Heronian triangle satisfy the equation, and any positive integer solution of the equation describes a Heronian triangle.
