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# Nine-point circle

In geometry, the nine-point circle is a circle that can be constructed for any given triangle. [1]

## Acute and obtuse triangles

An acute triangle is a triangle with all three angles acute (less than 90°).

## Altitude (triangle)

In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e. forming a right angle with) a line containing the base (the opposite side of the triangle).

## Angle

In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

## Apollonian circles

Apollonian circles are two families of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa.

## Barnes & Noble

Barnes & Noble, Inc., is a Fortune 500 company, the largest retail bookseller in the United States, and a leading retailer of content, digital media and educational products in the country.

## Charles Julien Brianchon

Charles Julien Brianchon (19 December 1783 &ndash; 29 April 1864) was a French mathematician and chemist.

## Circle

A circle is a simple shape in Euclidean geometry.

## Circumconic and inconic

In triangle geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.

## Circumscribed circle

In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon.

In mathematics, specifically projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting each pair of points.

## Concyclic points

In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle.

## Conic section

In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane.

## Cut-the-Knot

Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics.

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle.

## Ellipse

In mathematics, an ellipse is a curve on a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

## Euler line

In geometry, the Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral.

## Feuerbach point

In the geometry of triangles, the incircle and nine-point circle of a triangle are tangent to each other at the Feuerbach point of the triangle.

## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

## Holt McDougal

Holt McDougal is an American publishing company, a division of Houghton Mifflin Harcourt, that specializes in textbooks for use in secondary schools.

## Homothetic transformation

In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ. In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if) or reverse (if) the direction of all vectors.

## Incircle and excircles of a triangle

In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides.

## Jean-Victor Poncelet

Jean-Victor Poncelet (July 1, 1788 – December 22, 1867) was a French engineer and mathematician who served most notably as the commandant general of the École polytechnique.

## 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 (points where at least two of them meet): the common point H that they all share, and for each of the three pairs of circles one more intersection point (referred here as their 2-wise intersection).

## Karl Wilhelm Feuerbach

Karl Wilhelm von Feuerbach (30 May 1800 &ndash; 12 March 1834) was a German geometer and the son of legal scholar Paul Johann Anselm Ritter von Feuerbach, and the brother of philosopher Ludwig Feuerbach.

## Lester R. Ford

Lester Randolph Ford, Sr. (October 25, 1886, Missouri &ndash; November 11, 1967, Charlottesville, Virginia) was an American mathematician, editor of the American Mathematical Monthly from 1942 to 1946, and President of the Mathematical Association of America from 1947 to 1948.

## Lester's theorem

In Euclidean plane geometry, Lester's theorem, named after June Lester, states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle.

## Line segment

In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points.

## Medial triangle

The medial triangle or midpoint triangle of a triangle ABC is the triangle with vertices at the midpoints of the triangle's sides AB, AC and BC.

## Midpoint

In geometry, the midpoint is the middle point of a line segment.

## Napoleon points

In geometry, Napoleon points are a pair of special points associated with a plane triangle.

## Nine-point center

In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle.

## Nine-point hyperbola

In plane geometry with triangle ABC, the nine-point hyperbola is an instance of the nine-point conic described by Maxime Bôcher in 1892.

## Olry Terquem

Olry Terquem (16 June 1782 – 6 May 1862) was a French mathematician.

## Orthocentric system

In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three.

## Perpendicular

In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees).

## Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space.

## Polar circle (geometry)

In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is r^2 &.

## Poncelet point

In geometry, the Poncelet point of four given points is defined as follows: Let A, B, C, and D be four points in the plane that do not form an orthocentric system.

## Synthetic geometry

Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates.

## Tangent circles

In geometry, tangent circles (also known as kissing circles) are circles in a common plane that intersect in a single point.

## Tangential triangle

In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertices.

## Triangle

A triangle is a polygon with three edges and three vertices.

## Triangle center

In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles.

## Trilinear coordinates

In geometry, the trilinear coordinates x:y:z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle.

## Vertex (geometry)

In geometry, a vertex (plural vertices) is a special kind of point that describes the corners or intersections of geometric shapes.

## References

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