Inscribed angle

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In geometry, an inscribed angle is formed when two secant lines of a circle (or, in a degenerate case, when one secant line and one tangent line of that circle) intersect on the circle.

Typically, it is easiest to think of an inscribed angle as being defined by two chords of the circle sharing an endpoint.

The basic properties of inscribed angles are discussed in Book 3, Propositions 20–22 of Euclid's Elements. These are the inscribed angle is half the central angle, inscribed angles on the same arc of a chord are equal and the sum of the two distinct inscribed angles of a chord is 180°.

Property[edit]

An inscribed angle is said to intersect an arc on the circle. The arc is the portion of the circle that is in the interior of the angle. The measure of the intercepted arc (equal to its central angle) is exactly twice the measure of the inscribed angle.

This single property has a number of consequences within the circle. For example, it allows one to prove that when two chords intersect in a circle, the products of the lengths of their pieces are equal. It also allows one to prove that the opposite angles of a cyclic quadrilateral are supplementary.

Proof[edit]

To understand this proof, it is useful to draw a diagram.

Inscribed angles where one chord is a diameter[edit]

InscribedAngle 1ChordDiam.svg

Let O be the center of a circle. 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 is point V and whose sides pass through points A and B.

Draw line OA. Angle BOA is a central angle; call it θ. Lines OV and OA are both radii of the circle, so they have equal lengths. Therefore triangle VOA is isosceles, so angle BVA (the inscribed angle) and angle VAO are equal; let each of them be denoted as ψ.

Angles BOA and AOV are supplementary. They 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 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° 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[edit]

InscribedAngle CenterCircle.svg

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 \,

therefore, by equation (1),

 \theta_0 = 2 \psi_0. \,

Inscribed angles with the center of the circle in their exterior[edit]

InscribedAngle CenterCircleExtV2.svg

[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 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 EVC - \angle DVE .

then let

 \psi_0 = \angle DVC,
 \psi_1 = \angle DVE,
 \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 DOE and EOC, and

 \angle DOC = \angle EOC - \angle DOE.

Let

 \theta_0 = \angle DOC,
 \theta_1 = \angle DOE,
 \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.

Theorem[edit]

The inscribed angle θ is half of the central angle 2θ that subtends the same arc on the circle (magenta). Thus, the angle θ does not change as its vertex is moved around on the circle (green, blue and gold angles).
Given the two points A and B, the set of points M in the plane for which the angle AMB is equal to α is an arc of a circle. The measure of the angle AOB, which O is the center of the circle, is 2α.

The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the circle.

The inscribed angle theorem is used in many proofs of elementary Euclidean geometry of the plane. A special case of the theorem is Thales' theorem, which states that the angle subtended by a diameter is always 90°, i.e., a right angle. As a consequence of the theorem, opposite angles of cyclic quadrilaterals 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 with respect to a circle.

Proof[edit]

In the simplest case, one leg of the inscribed angle is a diameter of the circle, i.e., passes through the center of the circle. Since that leg is a straight line, the supplement of the central angle equals 180° − 2θ. Drawing a segment from the center of the circle to the other point of intersection of the inscribed angle produces an isosceles triangle, made from two radii of the circle and the second leg of the inscribed angle. Since two of the angles in an isosceles triangle are equal and since the angles in a triangle must add up to 180°, it follows that the inscribed angle equals θ, half of the central angle.

This result may be extended to an arbitrarily inscribed angle by drawing a diameter from the vertex of the angle. This converts the general problem into two sub-cases in which a diameter is a leg of each angle. The arbitrary angle equals half of the sum of the two central angles that share the diameter as a leg. Adding the two subangles again yields the result that the inscribed angle is half of the central angle.

Note that the central angle for the golden inscribed angle is 360° − 2θ. Therefore, the half of it (and thus the measure of the golden inscribed angle) is 180° − θ.

The set of all points (locus) for which a line segment can be seen at angle measured θ contains two arcs (one of each side of the line segment with central angle 2θ). In the special case of 90°, there is exactly one circle with center the middle of the line segment.

Corollaries[edit]

By a similar argument, the angle between a chord and the tangent line at one of its intersection points equals half of the central angle subtended by the chord. See also Tangent lines to circles.

References[edit]

  • Ogilvy, C. S. (1990). Excursions in Geometry. Dover. pp. 17–23. ISBN 0-486-26530-7. 
  • Gellert W, Küstner H, Hellwich M, Kästner H (1977). The VNR Concise Encyclopedia of Mathematics. New York: Van Nostrand Reinhold. p. 172. ISBN 0-442-22646-2. 

External links[edit]