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Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint. The inscribed angle theorem relates the measure of an inscribed angle to that of the central angle intercepting the same arc. The inscribed angle theorem appears as Proposition 20 in Book 3 of Euclid's Elements.
The lune of Hippocrates is the upper left shaded area. It has the same area as the lower right shaded triangle. In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle.
If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (⌢ and ⌢). That is, 2 ∠ C A B = ∠ D O E − ∠ B O C {\displaystyle 2\angle {CAB}=\angle {DOE}-\angle {BOC}} , where O is the centre of the circle (secant–secant ...
Thales's theorem can also be used to find the centre of a circle using an object with a right angle, such as a set square or rectangular sheet of paper larger than the circle. [7] The angle is placed anywhere on its circumference (figure 1). The intersections of the two sides with the circumference define a diameter (figure 2).
A circular sector is shaded in green. Its curved boundary of length L is a circular arc. A circular arc is the arc of a circle between a pair of distinct points.If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than π radians (180 degrees); and the other arc, the major arc, subtends an angle ...
Conversely, a convex quadrilateral in which the four angle bisectors meet at a point must be tangential and the common point is the incenter. [4] According to the Pitot theorem, the two pairs of opposite sides in a tangential quadrilateral add up to the same total length, which equals the semiperimeter s of the quadrilateral:
Spieker circle – Inscribed circle of a triangle's medial triangle; Tangent circles – Circles related to a point in the plane; Taylor circle – Special circle in geometry; Twin circles – Two congruent circles within an arbelos; Unit circle – Circle with radius of one; Van Lamoen circle – Circle associated with any given triangle
Angle AOB is a central angle. A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). [1]