In geometry, the perpendicular bisector construction of a quadrilateral is a construction which produces a new quadrilateral from a given quadrilateral using the perpendicular bisectors to the sides of the former quadrilateral. This construction arises naturally in an attempt to find a replacement for the circumcenter of a quadrilateral in the case that is non-cyclic.
Suppose that the vertices of the quadrilateral are given by . Let be the perpendicular bisectors of sides respectively. Then their intersections , with subscripts considered modulo 4, form the consequent quadrilateral . The construction is then iterated on to produce and so on.
An equivalent construction can be obtained by letting the vertices of be the circumcenters of the 4 triangles formed by selecting combinations of 3 vertices of .
1. If is not cyclic, then is not degenerate. [1]
2. Quadrilateral is never cyclic. [1] Combining #1 and #2, is always nondegenrate.
3. Quadrilaterals and are homothetic, and in particular, similar. [2] Quadrilaterals and are also homothetic.
3. The perpendicular bisector construction can be reversed via isogonal conjugation. [3] That is, given , it is possible to construct .
4. Let be the angles of . For every , the ratio of areas of and is given by [3]
5. If is convex then the sequence of quadrilaterals converges to the isoptic point of , which is also the isoptic point for every . Similarly, if is concave, then the sequence obtained by reversing the construction converges to the Isoptic Point of the 's. [3]
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 Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD.
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector.
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 plane Euclidean geometry, a rhombus is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.
In geometry, the Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
In geometry, a set of points are said to be concyclic if they lie on a common circle. All concyclic points are at the same distance from the center of the circle. Three points in the plane that do not all fall on a straight line are concyclic, but four or more such points in the plane are not necessarily concyclic.
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Lines in a plane or higher-dimensional space are said to be concurrent if they intersect at a single point. They are in contrast to parallel lines.
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, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician.
In Euclidean geometry, a tangential quadrilateral or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons.
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 the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and circumscribed quadrilateral. It has also rarely been called a double circle quadrilateral and double scribed quadrilateral.
In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.
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 its excircle, its radius the exradius and its center the excenter. The excenter lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect. The ex-tangential quadrilateral is closely related to the tangential quadrilateral.
In geometry, the circumcenter of mass is a center associated with a polygon which shares many of the properties of the center of mass. More generally, the circumcenter of mass may be defined for simplicial polytopes and also in the spherical and hyperbolic geometries.
