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the midpoint polygon of a polygon P is the polygon whose vertices are the midpoints of the edges of P. It is sometimes called the Kasner polygon after
Midpoint_polygon
Point on a line segment which is equidistant from both endpoints
the midpoint of a diagonal between opposite vertices is the polygon's center. The midpoint-stretching polygon of a cyclic polygon P (a polygon whose
Midpoint
In geometry, the midpoint-stretching polygon of a cyclic polygon P is another cyclic polygon inscribed in the same circle, the polygon whose vertices are
Midpoint-stretching_polygon
Triangle with vertices at midpoints of another triangle's sides
triangle's sides AB, AC, and BC. It is the n = 3 case of the midpoint polygon of a polygon with n sides. The medial triangle is not the same thing as the
Medial_triangle
Equiangular and equilateral polygon
is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon. A regular n-sided polygon can be constructed with compass and
Regular_polygon
Path that surrounds an area
} An equilateral polygon is a polygon which has all sides of the same length (for example, a rhombus is a 4-sided equilateral polygon). To calculate the
Perimeter
Shape with three sides
trigon is a polygon consisting of three sides connected at three corners. It is one of the basic shapes in geometry and the simplest of polygons. The corners
Triangle
Segment from the center of a polygon to the midpoint of one of its sides
regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that
Apothem
Polygon shape with eight sides
polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of
Octagon
Theorem in geometry
is planar or not. The theorem can be generalized to the midpoint polygon of an arbitrary polygon. Referring to the diagram above, triangles ADC and HDG
Varignon's_theorem
Topics referred to by the same term
circumscribed circle) A midpoint polygon of another polygon This disambiguation page lists articles associated with the title Inscribed polygon. If an internal
Inscribed_polygon
Concept in geometry
the center to the midpoint of a side of the polygon; its length, h, is less than the circle radius. Also, let each side of the polygon have length s; then
Area_of_a_circle
Non-planar polygon with infinitely many sides
infinite skew polygons with vertices alternating between two parallel lines. Infinite helical polygons are 3-dimensional infinite skew polygons with vertices
Infinite_skew_polygon
Diagram showing applied forces and moments on a physical body
the applied forces are arranged as the edges of a polygon of forces or force polygon (see § Polygon of forces). A body is said to be "free" when it is
Free_body_diagram
Points on a common circle
cocyclic) if they lie on a common circle. A polygon whose vertices are concyclic is called a cyclic polygon, and the circle is called its circumscribing
Concyclic_points
Simple curve of Euclidean geometry
two possible arcs determined by the endpoints of a diameter, taking its midpoint as centre. In non-technical common usage it may mean the interior of the
Circle
Theorem about the midpoint of a line connecting squares on two sides of a triangle
the circumcircles of the polygons, which are diametrically opposed of the common vertex C {\textstyle C} . Then, the midpoint of the line segment D 1 D
Bottema's_theorem
Part of a line that is bounded by two distinct end points; line with two endpoints
segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices
Line_segment
Four-sided polygon
In geometry, a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words
Quadrilateral
Polygon associated with a compact Riemann surface
But this is the midpoint of the line segment joining two interior points of edges and hence lies in C, the interior of the polygon. This again contradicts
Fundamental_polygon
Shape with five sides
Greek πέντε (pente) 'five' and γωνία (gonia) 'angle') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
Pentagon
Property of objects which are scaled or mirrored versions of each other
similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. Two congruent shapes
Similarity_(geometry)
from the points on the surface, and the centre of a line segment is the midpoint of the two ends. For objects with several symmetries, the centre of symmetry
Centre_(geometry)
Circle that passes through the vertices of a triangle
generally, an n-sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon, or in the special
Circumcircle
Quadrilateral with sides of equal length
rhombus include diamond, lozenge, and calisson. Every rhombus is a simple polygon (having no self-intersections). A rhombus is a special case of a parallelogram
Rhombus
Quadrilateral with four right angles
180°). The dual polygon of a rectangle is a rhombus, as shown in the table below. The figure formed by joining, in order, the midpoints of the sides of
Rectangle
Polygon with 30 edges
thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees. The regular triacontagon is a constructible polygon, by an edge-bisection
Triacontagon
Mean position of all the points in a shape
three medians of the triangle (each median connecting a vertex with the midpoint of the opposite side). For other properties of a triangle's centroid, see
Centroid
Method of drawing geometric objects
same area as a given polygon, and regular polygons of 3, 4, or 5 sides (or one with twice the number of sides of a given polygon). But they could not
Straightedge and compass construction
Straightedge_and_compass_construction
Shape with six sides
the midpoints of the segments connecting the centroids of opposite triangles form another equilateral triangle. A skew hexagon is a skew polygon with
Hexagon
Algorithm for finding a zero of a function
{\displaystyle c} , the midpoint of the interval, c = a + b 2 {\displaystyle c={\frac {a+b}{2}}} ; Calculate the function value at the midpoint, f ( c ) {\displaystyle
Bisection_method
Shape with three equal sides
with three sides of equal length and three equal angles. It is a regular polygon, occasionally known as the regular triangle. It is a special case of an
Equilateral_triangle
Conversion of a vector-graphics image to a raster image
rasterize lines. Algorithms such as the midpoint circle algorithm are used to render circles onto a pixelated canvas. Polygons are a common representation of digital
Rasterisation
Lines which intersect at a single point
to the midpoint of the opposite side. The three medians meet at the centroid. Perpendicular bisectors are lines running out of the midpoints of each
Concurrent_lines
Triangle with at least two sides congruent
angle bisector from the apex to the base, the median from the apex to the midpoint of the base, the perpendicular bisector of the base within the triangle
Isosceles_triangle
2024 film by Jane Schoenbrun
2025. Zilko, Christian (June 19, 2024). "3 Lessons We've Learned at the Midpoint of a Strange Year for Horror Movies". IndieWire. Retrieved November 10
I_Saw_the_TV_Glow
Non-convex polygons can be made convex by flips
making non-convex polygons convex that has also been studied is to perform flipturns, 180-degree rotations of a pocket around the midpoint of its convex hull
Erdős–Nagy_theorem
Flat-sided three-dimensional shape
ἕδρον (-hedron) 'base, seat') is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron"
Polyhedron
Summation method for divergent series
consequently the Borel polygon Π A {\displaystyle \Pi _{A}} is given by the regular m-gon centred at the origin, and such that 1 ∈ C is a midpoint of an edge. The
Borel_summation
Operation in Euclidean geometry
complete-truncation, is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting
Rectification_(geometry)
Division of something into two equal or congruent parts
of bisectors are the segment bisector, a line that passes through the midpoint of a given segment, and the angle bisector, a line that passes through
Bisection
Operation that cuts polytope vertices, creating a new facet in place of each vertex
beyond the midpoint of the edges, causing self-intersecting star polyhedra, and can parametrically relate to some of the regular star polygons and uniform
Truncation_(geometry)
11-pointed star polygon
In geometry, a hendecagram (also endecagram or endekagram) is a star polygon that has eleven vertices. The name hendecagram combines a Greek numeral prefix
Hendecagram
Line-drawing algorithm
of computer graphics. An extension to the original algorithm called the midpoint circle algorithm may be used for drawing circles. While algorithms such
Bresenham's_line_algorithm
Five tiles used in Islamic decorative art
An interlocking decagram-polygon mosaic design An interlocking decagram-polygon mosaic design An interlocking decagram-polygon mosaic design First, divide
Girih_tile
Shape with four equal sides and angles
equivalent ways. If a polygon in the Euclidean plane satisfies any one of the following criteria, it satisfies all of them: A square is a polygon with four equal
Square
Algorithms for mesh generation
unstructured grid. The computer uses a meshing algorithm to convert the polygonal model into triangles suitable for the finite element method. Chew's second
Delaunay_refinement
Half of the sum of side lengths of a polygon
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears
Semiperimeter
Rule from the theory of the tiling of the plane
each one is congruent to itself when rotated by 180-degrees around its midpoint. some of the six points may coincide but at least three of them must be
Conway_criterion
Geometric figure which is "snugly enclosed" by another figure
polygons, and triangles or regular polygons inscribed in circles. A circle inscribed in any polygon is called its incircle, in which case the polygon
Inscribed_figure
Size of a two-dimensional surface
synecdoche, "area" sometimes is used to refer to the region, as in a "polygonal area". The area of a shape can be measured by comparing the shape to squares
Area
Fundamental unit of a texture map
used to define their spatial relationships—divisions are made at the midpoints between the centroids of each texel and the centroids of every surrounding
Texel_(graphics)
Polyhedron with two kinds of faces
truncating the vertices fully, until each original edge is reduced to its midpoint. This sequence continues as the trihexagonal tiling, vertex figure (3.6)2
Quasiregular_polyhedron
Convex polygon which can tile the plane by itself
a regular polygon). Thus, the edges of k-dual uniform tilings coincide with centroid-to-edge-midpoint line segments of all regular polygons in the k-uniform
Planigon
orbit. When P is a polygon, some points might not have well-defined orbits, on account of the potential ambiguity of choosing the midpoint of the relevant
Outer_billiards
Group of symmetries of a regular polygon
mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the
Dihedral_group
Tiling hobbyist
that helped to solve the einstein problem. Smith discovered a 13-sided polygon in November 2022 whilst using a software package called PolyForm Puzzle
David Smith (amateur mathematician)
David_Smith_(amateur_mathematician)
Texture mapping technique
implemented by displacing the texture coordinates at a point on the rendered polygon by a function of the view angle in tangent space (the angle relative to
Parallax_mapping
Circle associated with a quadratic equation
circles have been used to develop ruler-and-compass constructions of regular polygons. Given a quadratic equation in the form x2 − sx + p = 0 the circle associated
Carlyle_circle
Generalization of a polytope in real space
polygon is a truncation of a regular polygon. A quasiregular polygon contains alternate edges of the regular polygons and . The quasiregular polygon has
Complex_polytope
Covering by shapes without overlaps or gaps
repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles
Tessellation
Relationship between two lines that meet at a right angle
maltitudes of a quadrilateral is a perpendicular to a side through the midpoint of the opposite side. An orthodiagonal quadrilateral is a quadrilateral
Perpendicular
Fractal creation method
chaos game originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created
Chaos_game
Geometric shape
angles. The equation of a semicircle with radius r {\displaystyle r} and midpoint ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} on the diameter between its
Semicircle
Approach to finding numerical solutions of ordinary differential equations
methods, such as the midpoint method also illustrated in the figures, behave more favourably: the global error of the midpoint method is roughly proportional
Euler_method
Construction on any polygon that yields a regular polygon with the same number of sides
arbitrary planar polygons. The theorem asserts that a certain procedure when applied to an arbitrary polygon always yields a regular polygon having the same
Petr–Douglas–Neumann_theorem
Line constructed from a triangle
hypotenuse—that is, it goes through both the right-angled vertex and the midpoint of the side opposite that vertex. This is because the right triangle's
Euler_line
Quadrilateral symmetric across a diagonal
into two pairs of adjacent equal-length sides. One diagonal crosses the midpoint of the other diagonal at a right angle, forming its perpendicular bisector
Kite_(geometry)
Point not between two other points
or a proper interval if its endpoints are distinct. The midpoint of an interval is the midpoint of its endpoints. The closed interval [ x , y ] {\displaystyle
Extreme_point
Shape formed from points common to other shapes
the polygons into small sub-polygons and determines the smallest window (rectangle with sides parallel to the coordinate axes) for any sub-polygon. Before
Intersection_(geometry)
Mathematical treatise by Euclid
there are infinitely many prime numbers, and the construction of regular polygons and polyhedra. Often referred to as the most successful textbook ever written
Euclid's_Elements
Quadrilateral with two pairs of parallel sides
of the vector cross product of two adjacent sides. Any line through the midpoint of a parallelogram bisects the area. Any non-degenerate affine transformation
Parallelogram
Polygon with four crossed edges of two lengths
discovered, if an antiparallelogram has its long side fixed in this way, the midpoint of the unfixed long edge will trace out a lemniscate or figure eight curve
Antiparallelogram
Near-cylindrical polyhedron with large area
circumscribed regular polygons. More generally, the length of any smooth or rectifiable curve can be defined as the supremum of the lengths of polygonal chains inscribed
Schwarz_lantern
Center of the inscribed circle of a triangle
group of triangle centers. For polygons with more than three sides, the incenter only exists for tangential polygons: those that have an incircle that
Incenter
Universality of construction using just a straightedge and a single circle with center
such that B is the midpoint: Draw an arbitrary line (in red) passing through the given circle's center, O, and the desired midpoint B (chosen arbitrarily)
Poncelet–Steiner_theorem
2012 video game
early status as a "victim". Yohalem added that Vaas's death at the game's midpoint was inspired by the novel To the Lighthouse, in which the protagonist died
Far_Cry_3
Rectangle with side lengths in the golden ratio
straightedge and compass in four steps: Draw a square Draw a line from the midpoint of one side of the square to an opposite corner Use that line as the radius
Golden_rectangle
Season of television series
Breaux & Jake Connelly Join Cast; BTS Video Marks Final Season Filming Midpoint". Deadline Hollywood. Archived from the original on July 15, 2024. Retrieved
Stranger_Things_season_5
Surveyed border line between U.S. states of Delaware, Maryland, and Pennsylvania
is not a true straight line in the geometric sense, but is instead a polygonal chain, a series of many adjoining line segments, following a path between
Mason–Dixon_line
Curve used in computer graphics and related fields
polygon formed by connecting the Bézier points with lines, starting with P0 and finishing with Pn, is called the Bézier polygon (or control polygon)
Bézier_curve
Any of the five regular polyhedra
means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of
Platonic_solid
Shape made by slicing off a corner of a polytope
When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can
Vertex_figure
Polygon whose four sides all touch a circle
quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons. Other less frequently used names for this class of quadrilaterals are
Tangential_quadrilateral
Dynamical system involving reflection
all polygons have periodic orbits. In this model, a light ray with a given initial position and orientation follows a path in the form of a polygonal chain
Triangular_billiards
Four-dimensional analog of the dodecahedron
every convex regular polytope in the first four dimensions (except the polygons {7} and above). As the sixth and largest regular convex 4-polytope, it
120-cell
Right triangle with a feature making calculations on the triangle easier
an equilateral triangle ABC with side length 2, and with point M as the midpoint of segment BC. Draw an altitude line from A to M. Then ABM is a 30°–60°–90°
Special_right_triangle
Surname list
Einstein's theory of general relativity Kasner polygon of a polygon P is the polygon whose vertices are the midpoints of the edges of P Kasner's dwarf burrowing
Kasner
Quadrilateral with equal perpendicular diagonals
midsquare quadrilateral (referring to the square formed by its four edge midpoints). These shapes are, by definition, simultaneously equidiagonal quadrilaterals
Midsquare_quadrilateral
Circles tangent to all three sides of a triangle
inellipse – Unique ellipse tangent to all 3 midpoints of a given triangle's sides Tangential quadrilateral – Polygon whose four sides all touch a circle Triangle
Incircle_and_excircles
Quadrilateral whose vertices lie on a circle
quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertices all lie on a single circle, making the sides chords of
Cyclic_quadrilateral
2016 video game
elaborated that the ending, which focused on Marchenko, was intended as the midpoint and the game would have seen Jensen travelling to the city of Rabi'ah and
Deus_Ex:_Mankind_Divided
Line which touches a circle at exactly one point
a point P external to the circle C: A circle is drawn centered on the midpoint M of the line segment OP, having diameter OP, where O is again the center
Tangent_lines_to_circles
Polyhedron with four faces
tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the
Tetrahedron
League of Legends esports team
practice regimens proved to be a challenging adjustment for Chaox. By the midpoint of the season, in February 2013, the team signed Jason "WildTurtle" Tran
TSM League of Legends division
TSM_League_of_Legends_division
Straight line segment that passes through the centre of a circle
straightedge and compass, by finding the midpoint of the segment and then drawing the circle centered at the midpoint through one of the ends of the line segment
Diameter
Method to evaluate polynomials in Bernstein form
* t = a + (b - a) * t midpoints.push([ ax + (bx - ax) * t, ay + (by - ay) * t, ]); } retArr.push (midpoints) points = midpoints; } return retArr; } For
De_Casteljau's_algorithm
Line constructed from a triangle
reflection of the foot of the altitude (dropped onto the side line) about the midpoint of the side line being constructed. Furthermore, this point is a tangent
Simson_line
Crater on the Moon
the worn outer walls, and a complex of central hills to the north of the midpoint. The crater Doppler B lies within the northern rim, and occupies a portion
Doppler_(crater)
2019 American superhero drama television series
Being", Lindelof felt the story was closer to its ending rather than as a midpoint, and that if they continued for four additional episodes, one of them would
Watchmen_(TV_series)
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