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. Usually, the radius is required to be a positive number. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.
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 quadri, a variant of four, and latus, meaning "side". Another name for it is tetragon, derived from greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to e.g., pentagon. "Gon" being "angle" also is at the root of calling it quadrangle, 4-angle, in analogy to triangle. A quadrilateral with vertices , , and is sometimes denoted as .
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle.
In geometry, Thales' 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 projective geometry, Desargues's theorem, named after Girard Desargues, states:
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, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.
In geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible. It is so named because this problem was first raised by Fermat in a private letter to Evangelista Torricelli, who solved it.
Napoleon's problem is a compass construction problem. In it, a circle and its center are given. The challenge is to divide the circle into four equal arcs using only a compass. Napoleon was known to be an amateur mathematician, but it is not known if he either created or solved the problem. Napoleon's friend the Italian mathematician Lorenzo Mascheroni introduced the limitation of using only a compass into geometric constructions. But actually, the challenge above is easier than the real Napoleon's problem, consisting in finding the center of a given circle with compass alone. The following sections will describe solutions to three problems and proofs that they work.
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, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them. Its essential role was discovered by Moritz Pasch in 1882.
In geometry, the Desargues configuration is a configuration of ten points and ten lines, with three points per line and three lines per point. It is named after Girard Desargues, and closely related to Desargues' theorem, which proves the existence of the configuration.
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem is an important theorem in elementary geometry about the ratios of various line segments that are created if two intersecting lines are intercepted by a pair of parallels. It is equivalent to the theorem about ratios in similar triangles. Traditionally it is attributed to Greek mathematician Thales. It was known to the ancient Babylonians and Egyptians although its first known proof appears in Euclid's Elements.
In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states
In mathematics, the Pythagorean theorem, or Pythagoras' 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:
In Euclidean geometry, the Droz-Farny line theorem is a property of two perpendicular lines through the orthocenter of an arbitrary triangle.
Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed. The fifth axiom of hyperbolic geometry says that given a line L and a point P not on that line, there are at least two lines passing through P that are parallel to L. As in Euclidean geometry, where ancient Greek mathematicians used a compass and idealized ruler for constructions of lengths, angles, and other geometric figures, constructions can also be made in hyperbolic geometry.
The theorem of the gnomon states that certain parallelograms occurring in a gnomon have areas of equal size.
Maxwell's theorem is the following statement about triangles in the plane.
For a given triangle and a point not on the sides of that triangle construct a second triangle , such that the side is parallel to the line segment , the side is parallel to the line segment and the side is parallel to the line segment . Then the parallel to through , the parallel to through and the parallel to through intersect in a common point .
