A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling, possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other.
In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects.
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the opposite side is called the extended base of the altitude. The intersection of the extended base and the altitude is called the foot of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude at that vertex. It is a special case of orthogonal projection.
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.
In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.
In geometry, symmedians are three particular geometrical lines associated with every triangle. They are constructed by taking a median of the triangle, and reflecting the line over the corresponding angle bisector. The angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector.
In projective geometry, Pascal's theorem states that if six arbitrary points are chosen on a conic and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon meet at three points which lie on a straight line, called the Pascal line of the hexagon. It is named after Blaise Pascal.
In mathematics, Pappus's hexagon theorem states that
In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. The line through these points is the Simson line of P, named for Robert Simson. The concept was first published, however, by William Wallace in 1799.
In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle points , , and lie on segments , , and , then writing , , and , the signed area of the triangle formed by the cevians , , and is the area of triangle times
In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician.
In Euclidean geometry, the Erdős–Mordell inequality states that for any triangle ABC and point P inside ABC, the sum of the distances from P to the sides is less than or equal to half of the sum of the distances from P to the vertices. It is named after Paul Erdős and Louis Mordell. Erdős (1935) posed the problem of proving the inequality; a proof was provided two years later by Mordell and D. F. Barrow (1937). This solution was however not very elementary. Subsequent simpler proofs were then found by Kazarinoff (1957), Bankoff (1958), and Alsina & Nelsen (2007).
Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram, from an arbitrary quadrilateral (quadrangle). It is named after Pierre Varignon, whose proof was published posthumously in 1731.
Miquel's theorem is a result in geometry, named after Auguste Miquel, concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides. It is one of several results concerning circles in Euclidean geometry due to Miquel, whose work was published in Liouville's newly founded journal Journal de mathématiques pures et appliquées.
In mathematics, the Pythagorean theorem, or Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:
