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Circles of Apollonius

The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for exampl ...

Brocard circle

In geometry, the Brocard circle for a triangle is a circle defined from a given triangle. It passes through the circumcenter and symmedian of the triangle, and is centered at the midpoint of the line segment joining them.

Circular sector

A circular sector or circle sector, is the portion of a disk 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 in radians, r {\displ ...

Circular segment

In geometry, a circular segment is a region of a circle which is "cut off" from the rest of the circle by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is bounded by an arc of a circle and by the ...

Extouch triangle

The vertices of the extouch triangle are given in trilinear coordinates by: T A = 0: csc 2 ⁡ B / 2: csc 2 ⁡ C / 2 {\displaystyle T_{A}=0:\csc ^{2}{\leftB/2\right}:\csc ^{2}{\leftC/2\right}} T B = csc 2 ⁡ A / 2: 0: csc 2 ⁡ C / 2 {\displaystyle T_{ ...

Fuhrmann circle

In geometry, the Fuhrmann circle of a triangle, named after the German Wilhelm Fuhrmann, is the circle with a diameter of the line segment between the orthocenter H {\displaystyle H} and the Nagel point N {\displaystyle N}. This circle is identic ...

Johnson circles

In geometry, a set of Johnson circles comprises three circles of equal radius r sharing one common point of intersection H. In such a configuration the circles usually have a total of four intersections: the common point H that they all share, an ...

Malfatti circles

In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem of ...

Schinzel circle

Schinzel circles are a set of circles with a given number of integer points on the circumference of the circle. If the number n of points on the circumference of the circle is even, n = 2 k, then a Schinzel circle is given by: x − 1 2 + y 2 = 1 4 ...

Spieker circle

In geometry, the incircle of the medial triangle of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker. Its center, the Spieker center, in addition to being the incenter of the medial triangle, is the cente ...

Tangent circles

In geometry, tangent circles are circles in a common plane that intersect in a single point. There are two types of tangency: internal and external. Many problems and constructions in geometry are related to tangent circles; such problems often h ...

Tangent lines to circles

In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circles interior. Tangent lines to circles form the subject of several theorems, and play an important role in many ...

Van Lamoen circle

In Euclidean plane geometry, the van Lamoen circle is a special circle associated with any given triangle T {\displaystyle T}. It contains the circumcenters of the six triangles that are defined inside T {\displaystyle T} by its three medians. Sp ...

Prince Ruperts cube

In geometry, Prince Ruperts cube is the largest cube that can pass through a hole cut through a unit cube, i.e. through a cube whose sides have length 1, without splitting the cube into two pieces. Its side length is approximately 6% larger than ...

5-Con triangles

In geometry, two triangles are said to be 5-Con or almost congruent if they are not congruent triangles but they are similar triangles and share two side lengths. The 5-Con triangles are important examples for understanding the solution of triang ...

Calabi triangle

The Calabi triangle is a special triangle found by Eugenio Calabi and defined by its property of having three different placements for the largest square that it contains. It is an obtuse isosceles triangle with an irrational but algebraic ratio ...

Right triangle

A right triangle or right-angled triangle is a triangle in which one angle is a right angle. The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the hypotenuse s ...

Archimedean circle

In geometry, an Archimedean circle is any circle constructed from an arbelos that has the same radius as each of Archimedes twin circles. The radius ρ of such a circle is given by ρ = 1 2 r 1 − r, {\displaystyle \rho ={\frac {1}{2}}r\left1-r\righ ...

Archimedes quadruplets

In geometry, Archimedes quadruplets are four congruent circles associated with an arbelos. Introduced by Frank Power in the summer of 1998, each have the same area as Archimedes twin circles, making them Archimedean circles.

Schoch circles

In 1979, Thomas Schoch discovered a dozen new Archimedean circles; he sent his discoveries to Scientific American s "Mathematical Games" editor Martin Gardner. The manuscript was forwarded to Leon Bankoff. Bankoff gave a copy of the manuscript to ...

Schoch line

In geometry, the Schoch line is a line defined from an arbelos and named by Peter Woo after Thomas Schoch, who had studied it in conjunction with the Schoch circles.

Twin circles

In geometry, the twin circles are two special circles associated with an arbelos. An arbelos is determined by three collinear points A, B, and C, and is the curvilinear triangular region between the three semicircles that have AB, BC, and AC as t ...

Straightedge and compass construction

Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass. The idealized ruler, kno ...

Compass equivalence theorem

The compass equivalence theorem is an important statement in compass and straightedge constructions. The tool advocated by Plato in these constructions is a divider or collapsing compass, that is, a compass that "collapses" whenever it is lifted ...

360-gon

A regular 360-gon is represented by Schlafli symbol {360} and also can be constructed as a truncated 180-gon, t{180}, or a twice-truncated enneacontagon, tt{90}, or a thrice-truncated tetracontapentagon, ttt{45}. One interior angle in a regular 3 ...

Apothem

The apothem of a 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 is perpendicular to one of its sides. The word "apothem" can also refer ...

Bicentric quadrilateral

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called inradius and circumradius, and incenter and circumcenter respectively. From ...

Concave polygon

A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle - that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive. Some l ...

Dual polygon

Regular polygons are self-dual. The dual of an isogonal vertex-transitive polygon is an isotoxal edge-transitive polygon. For example, the isogonal rectangle and isotoxal rhombus are duals. In a cyclic polygon, longer sides correspond to larger e ...

Enneacontagon

In geometry, an enneacontagon or enenecontagon or 90-gon is a ninety-sided polygon. The sum of any enneacontagons interior angles is 15840 degrees. A regular enneacontagon is represented by Schlafli symbol {90} and can be constructed as a truncat ...

Enneadecagon

As 19 is a Pierpont prime but not a Fermat prime, the regular enneadecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisector. Another animation of an approximate construction. ...

Equiangular polygon

In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon. Isogonal polygons are equiangular polygons which alternate two edge lengths.

Equilateral pentagon

In geometry an equilateral pentagon is a polygon with five sides of equal length. Its five internal angles, in turn, can take a range of sets of values, thus permitting it to form a family of pentagons. The requirement is that all angles must add ...

Equilateral polygon

In geometry, three or more than three straight lines make a polygon and an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, it doesn’t need to be equiangular, but if it does then it is a regula ...

Ex-tangential quadrilateral

In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside the quadrilateral. It has also been called an exscriptible quadrilateral. The circle is called ...

Golygon

A golygon is any polygon with all right angles whose sides are consecutive integer lengths. Golygons were invented and named by Lee Sallows, and popularized by A.K. Dewdney in a 1990 Scientific American column. Variations on the definition of gol ...

Hendecagon

A regular hendecagon is represented by Schlafli symbol {11}. A regular hendecagon has internal angles of 147. 27 degrees =147 3 11 {\displaystyle {\tfrac {3}{11}}} degrees. The area of a regular hendecagon with side length a is given by A = 11 4 ...

Heptacontagon

In geometry, a heptacontagon or 70-gon is a seventy-sided polygon. The sum of any heptacontagons interior angles is 12240 degrees. A regular heptacontagon is represented by Schlafli symbol {70} and can also be constructed as a truncated triaconta ...

Icosidigon

As 22 = 2 × 11, the icosidigon can be constructed by truncating a regular hendecagon. However, the icosidigon is not constructible with a compass and straightedge, since 11 is not a Fermat prime. Consequently, the icosidigon cannot be constructed ...

Icosihexagon

In geometry, an icosihexagon or 26-gon is a twenty-six-sided polygon. The sum of any icosihexagons interior angles is 4320 degrees.

Icosioctagon

In geometry, an icosioctagon or 28-gon is a twenty eight sided polygon. The sum of any icosioctagons interior angles is 4680 degrees.

Infinite skew polygon

In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope has vertices that are not all colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lines ...

Isosceles trapezoid

In Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined as a trapezoid in which both legs and both ...

Encyclopedic dictionary

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Pino - logical board game which is based on tactics and strategy. In general this is a remix of chess, checkers and corners. The game develops imagination, concentration, teaches how to solve tasks, plan their own actions and of course to think logically. It does not matter how much pieces you have, the main thing is how they are placement!

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