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The number of points (n), chords (c) and regions (r G) for first 6 terms of Moser's circle problem. In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes called Moser's circle problem (named after Leo Moser), has a solution by an inductive method.
The radius of each of the three Malfatti circles may be determined as a formula involving the three side lengths a, b, and c of the triangle, the inradius r, the semiperimeter = (+ +) /, and the three distances d, e, and f from the incenter of the triangle to the vertices opposite sides a, b, and c respectively.
For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: each of the two possible triangulations that split the quadrangle into two triangles satisfies the "Delaunay condition", i.e., the requirement that the circumcircles of all triangles have empty interiors.
Angle AOB is a central angle. A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). [1]
In all the formulas stated below the sides a, b, and c must be measured in absolute length, a unit so that the Gaussian curvature K of the plane is −1. In other words, the quantity R in the paragraph above is supposed to be equal to 1. Trigonometric formulas for hyperbolic triangles depend on the hyperbolic functions sinh, cosh, and tanh.
These include the equality of all the distances between vertices; the equality of all the distances from vertices to the center of the simplex; the fact that the angle subtended through the new vertex by any two previously chosen vertices is /; and the fact that the angle subtended through the center of the simplex by any two vertices is (/).
The defect of any of the vertices of a regular dodecahedron (in which three regular pentagons meet at each vertex) is 36°, or π/5 radians, or 1/10 of a circle. Each of the angles measures 108°; three of these meet at each vertex, so the defect is 360° − (108° + 108° + 108°) = 36°.
When the outer Soddy circle has positive curvature, both Soddy centers are equal detour points. When the outer Soddy circle has negative curvature, its center is the isoperimetric point: the triangles ABP 2, BCP 2, and CAP 2 have equal perimeter. In geometry, the Soddy circles of a triangle are two circles associated with any triangle in the plane.