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Equal chords are subtended by equal angles from the center of the circle. A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle. If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem).
For example, the first Napoleon point is the point of concurrency of the three lines each from a vertex to the centroid of the equilateral triangle drawn on the exterior of the opposite side from the vertex. A generalization of this notion is the Jacobi point. The de Longchamps point is the point of concurrence of several lines with the Euler line.
Any straight line segment connecting two points on a circle or ellipse is called a chord. Any chord in a circle which has no longer chord is called a diameter, and any segment connecting the circle's center (the midpoint of a diameter) to a point on the circle is called a radius.
The diameter is the longest chord of the circle. Among all the circles with a chord AB in common, the circle with minimal radius is the one with diameter AB. If the intersection of any two chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then ab = cd.
Examples of cyclic quadrilaterals. In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertices all lie on a single circle, making the sides chords of the circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.
Any line perpendicular to any chord of a circle and passing through its midpoint also passes through the circle's center. The butterfly theorem states that, if M is the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly, then M is the midpoint of XY.
The number of points (n), chords (c) and regions (r G) for first 6 terms of Moser's circle problem. In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes called Moser's circle problem (named after Leo Moser), has a solution by an inductive method.
In geometry, the sagitta (sometimes abbreviated as sag [1]) of a circular arc is the distance from the midpoint of the arc to the midpoint of its chord. [2] It is used extensively in architecture when calculating the arc necessary to span a certain height and distance and also in optics where it is used to find the depth of a spherical mirror ...