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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, an inscribed angle is the
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
formed in the interior of a
circle 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 cons ...
when two
chords Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord ( ...
intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. 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 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 le ...
subtending the same arc. The inscribed angle theorem appears as Proposition 20 on Book 3 of Euclid's ''Elements''.


Theorem


Statement

The inscribed angle theorem states that an angle ''θ'' inscribed in a circle is half of the central angle 2''θ'' that
subtend In geometry, an angle is subtended by an arc, line segment or any other section of a curve when its two rays pass through the endpoints of that arc, line segment or curve section. Conversely, the arc, line segment or curve section confined wi ...
s the same arc on the circle. Therefore, the angle does not change as its
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
is moved to different positions on the circle.


Proof


Inscribed angles where one chord is a diameter

Let ''O'' be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them ''V'' and ''A''. Draw line ''VO'' and extended past ''O'' so that it intersects the circle at point ''B'' which is diametrically opposite the point ''V''. Draw an angle whose
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
is point ''V'' and whose sides pass through points ''A'' and ''B''. Draw line ''OA''. Angle ''BOA'' 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 le ...
; call it ''θ''. Lines ''OV'' and ''OA'' are both
radii In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of the circle, so they have equal lengths. Therefore, triangle ''VOA'' is
isosceles In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
, so angle ''BVA'' (the inscribed angle) and angle ''VAO'' are equal; let each of them be denoted as ''ψ''. Angles ''BOA'' and ''AOV'' add up to 180°, since line ''VB'' passing through ''O'' is a straight line. Therefore, angle ''AOV'' measures 180° − ''θ''. It is known that the three angles of a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
add up to 180°, and the three angles of triangle ''VOA'' are: : 180° − ''θ'' : ''ψ'' : ''ψ''. Therefore, : 2 \psi + 180^\circ - \theta = 180^\circ. Subtract : (180^\circ - \theta) from both sides, : 2 \psi = \theta, where ''θ'' is the central angle subtending arc ''AB'' and ''ψ'' is the inscribed angle subtending arc ''AB''.


Inscribed angles with the center of the circle in their interior

Given a circle whose center is point ''O'', choose three points ''V'', ''C'', and ''D'' on the circle. Draw lines ''VC'' and ''VD'': angle ''DVC'' is an inscribed angle. Now draw line ''VO'' and extend it past point ''O'' so that it intersects the circle at point ''E''. Angle ''DVC'' subtends arc ''DC'' on the circle. Suppose this arc includes point ''E'' within it. Point ''E'' is diametrically opposite to point ''V''. Angles ''DVE'' and ''EVC'' are also inscribed angles, but both of these angles have one side which passes through the center of the circle, therefore the theorem from the above Part 1 can be applied to them. Therefore, : \angle DVC = \angle DVE + \angle EVC. then let : \psi_0 = \angle DVC, : \psi_1 = \angle DVE, : \psi_2 = \angle EVC, so that : \psi_0 = \psi_1 + \psi_2. \qquad \qquad (1) Draw lines ''OC'' and ''OD''. Angle ''DOC'' is a central angle, but so are angles ''DOE'' and ''EOC'', and : \angle DOC = \angle DOE + \angle EOC. Let : \theta_0 = \angle DOC, : \theta_1 = \angle DOE, : \theta_2 = \angle EOC, so that : \theta_0 = \theta_1 + \theta_2. \qquad \qquad (2) From Part One we know that \theta_1 = 2 \psi_1 and that \theta_2 = 2 \psi_2 . Combining these results with equation (2) yields : \theta_0 = 2 \psi_1 + 2 \psi_2 = 2(\psi_1 + \psi_2) therefore, by equation (1), : \theta_0 = 2 \psi_0.


Inscribed angles with the center of the circle in their exterior

The previous case can be extended to cover the case where the measure of the inscribed angle is the ''difference'' between two inscribed angles as discussed in the first part of this proof. Given a circle whose center is point ''O'', choose three points ''V'', ''C'', and ''D'' on the circle. Draw lines ''VC'' and ''VD'': angle ''DVC'' is an inscribed angle. Now draw line ''VO'' and extend it past point ''O'' so that it intersects the circle at point ''E''. Angle ''DVC'' subtends arc ''DC'' on the circle. Suppose this arc does not include point ''E'' within it. Point ''E'' is diametrically opposite to point ''V''. Angles ''EVD'' and ''EVC'' are also inscribed angles, but both of these angles have one side which passes through the center of the circle, therefore the theorem from the above Part 1 can be applied to them. Therefore, : \angle DVC = \angle EVC - \angle EVD . then let : \psi_0 = \angle DVC, : \psi_1 = \angle EVD, : \psi_2 = \angle EVC, so that : \psi_0 = \psi_2 - \psi_1. \qquad \qquad (3) Draw lines ''OC'' and ''OD''. Angle ''DOC'' is a central angle, but so are angles ''EOD'' and ''EOC'', and : \angle DOC = \angle EOC - \angle EOD. Let : \theta_0 = \angle DOC, : \theta_1 = \angle EOD, : \theta_2 = \angle EOC, so that : \theta_0 = \theta_2 - \theta_1. \qquad \qquad (4) From Part One we know that \theta_1 = 2 \psi_1 and that \theta_2 = 2 \psi_2 . Combining these results with equation (4) yields : \theta_0 = 2 \psi_2 - 2 \psi_1 therefore, by equation (3), : \theta_0 = 2 \psi_0.


Corollary

By a similar argument, the angle between a chord and the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
line at one of its intersection points equals half of the central angle subtended by the chord. See also
Tangent lines to circles In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
.


Applications

The inscribed angle
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
is used in many proofs of elementary
Euclidean geometry of the plane Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
. A special case of the theorem is
Thales' theorem In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line 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 ...
, which states that the angle subtended by a
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
is always 90°, i.e., a right angle. As a consequence of the theorem, opposite angles of
cyclic quadrilateral 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 ''c ...
s sum to 180°; conversely, any quadrilateral for which this is true can be inscribed in a circle. As another example, the inscribed angle theorem is the basis for several theorems related to the
power of a point In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826. Specifically, the power \Pi(P) of a point P with respect ...
with respect to a circle. Further, it allows one to prove that when two chords intersect in a circle, the products of the lengths of their pieces are equal.


Inscribed angle theorems for ellipses, hyperbolas and parabolas

Inscribed angle theorems exist for ellipses, hyperbolas and parabolas, too. The essential differences are the measurements of an angle. (An angle is considered a pair of intersecting lines.) *
Ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
*
Hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
*
Parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...


References

* * *


External links

*
Relationship Between Central Angle and Inscribed Angle

Munching on Inscribed Angles
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Arc Central Angle
With interactive animation

With interactive animation

With interactive animation
At bookofproofs.org
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