Inscribed angle

In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords 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 subtending the same arc.

The inscribed angle theorem appears as Proposition 20 in Book 3 of Euclid's Elements.

Note that this theorem is not to be confused with the Angle bisector theorem, which also involves angle bisection (but of an angle of a triangle not inscribed in a circle).

Theorem

Statement

For fixed 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 $∠ AOB$ , where $O$ is the center of the circle, is $2 α$ .

The inscribed angle theorem states that an angle $θ$ inscribed in a circle is half of the central angle $2 θ$ 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.

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 $OV$ 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 $A, B$ .

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, summing to a straight angle (180°), so angle $∠ AOV$ measures $180° - θ$ .

The three angles of triangle $△ VOA$ must sum to $180°$ :

$(180^{\circ }-\theta )+\psi +\psi =180^{\circ }.$

Adding $\theta -180^{\circ }$ to both sides yields

$2\psi =\theta .$

Inscribed angles with the center of the circle in their interior

Case: Center interior to angle
   $ψ 0 = ∠ DVC, θ 0 = ∠ DOC$

   $ψ 1 = ∠ EVD, θ 1 = ∠ EOD$

   $ψ 2 = ∠ EVC, θ 2 = ∠ EOC$

Given a circle whose center is point $O$ , choose three points $V, C, D$ on the circle. Draw lines $VC$ and $VD$ : angle $∠ DVC$ is an inscribed angle. Now draw line $OV$ 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 $∠ DVE, ∠ EVC$ 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

${\begin{aligned}\psi _{0}&=\angle DVC,\\\psi _{1}&=\angle DVE,\\\psi _{2}&=\angle EVC,\end{aligned}}$

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

${\begin{aligned}\theta _{0}&=\angle DOC,\\\theta _{1}&=\angle DOE,\\\theta _{2}&=\angle EOC,\end{aligned}}$

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 $V, C, D$ on the circle. Draw lines $VC$ and $VD$ : angle $∠ DVC$ is an inscribed angle. Now draw line $OV$ 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 $∠ EVD, ∠ EVC$ 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

${\begin{aligned}\psi _{0}&=\angle DVC,\\\psi _{1}&=\angle EVD,\\\psi _{2}&=\angle EVC,\end{aligned}}$

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

${\begin{aligned}\theta _{0}&=\angle DOC,\\\theta _{1}&=\angle EOD,\\\theta _{2}&=\angle EOC,\end{aligned}}$

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 line at one of its intersection points equals half of the central angle subtended by the chord. See also Tangent lines to circles.

Applications

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. 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.)

References

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.
Moise, Edwin E. (1974). Elementary Geometry from an Advanced Standpoint (2nd ed.). Reading: Addison-Wesley. pp. 192–197. ISBN 0-201-04793-4.

External links

Weisstein, Eric W. "Inscribed Angle". MathWorld.
Relationship Between Central Angle and Inscribed Angle
Munching on Inscribed Angles at cut-the-knot
Arc Central Angle With interactive animation
Arc Peripheral (inscribed) Angle With interactive animation
Arc Central Angle Theorem With interactive animation
At bookofproofs.github.io