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About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices. [17]
Prehistoric people made stone circles and timber circles, and circular elements are common in petroglyphs and cave paintings. [2] Disc-shaped prehistoric artifacts include the Nebra sky disc and jade discs called Bi. A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle. Another proof of this result, which relies only on two chord properties given above, is as follows. Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of the chord, we know that it is a part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter is ( 2 r − x) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that ( 2 r − x) x = ( y / 2) 2. Solving for r, we find the required result. The Egyptian Rhind papyrus, dated to 1700 BCE, gives a method to find the area of a circle. The result corresponds to 256 / 81 (3.16049...) as an approximate value of π. [3] The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are:
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 ( D E ⌢ {\displaystyle {\overset {\frown }{DE}}} and B C ⌢ {\displaystyle {\overset {\frown }{BC}}} ). 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 theorem). 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. If AD is tangent to the circle at A and if AQ is a chord of the circle, then ∠ DAQ = 1 / 2arc( AQ).
A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to the circle. The simplest and most basic is the construction given the centre of the circle and a point on the circle. Place the fixed leg of the compass on the centre point, the movable leg on the point on the circle and rotate the compass. In every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle. [16]If two secants, AE and AD, also cut the circle at B and C respectively, then AC × AD = AB × AE (corollary of the chord theorem). The circle signifies many sacred and spiritual concepts, including unity, infinity, wholeness, the universe, divinity, balance, stability and perfection, among others. Such concepts have been conveyed in cultures worldwide through the use of symbols, for example, a compass, a halo, the vesica piscis and its derivatives (fish, eye, aureole, mandorla, etc.), the ouroboros, the Dharma wheel, a rainbow, mandalas, rose windows and so forth. [8] Magic circles are part of some traditions of Western esotericism. However, differences in worldview (beliefs and culture) had a great impact on artists’ perceptions. While some emphasised the circle's perimeter to demonstrate their democratic manifestation, others focused on its centre to symbolise the concept of cosmic unity. In mystical doctrines, the circle mainly symbolises the infinite and cyclical nature of existence, but in religious traditions it represents heavenly bodies and divine spirits. If two angles are inscribed on the same chord and on the same side of the chord, then they are equal. Among all the circles with a chord AB in common, the circle with minimal radius is the one with diameter AB.
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