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

The inscribed angle theorem states that an angle inscribed in a circle is half of the central angle 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 be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them and . Draw line and extended past so that it intersects the circle at point which is diametrically opposite the point . Draw an angle whose vertex is point and whose sides pass through points .

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

Angles and are supplementary, summing to a straight angle (180°), so angle measures .

The three angles of triangle must sum to :

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

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

$2\psi = \theta.$

Inscribed angles with the center of the circle in their interior

Given a circle whose center is point , choose three points on the circle. Draw lines and : angle is an inscribed angle. Now draw line and extend it past point so that it intersects the circle at point . Angle subtends arc on the circle.

Suppose this arc includes point within it. Point is diametrically opposite to point . Angles 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

$\begin \psi_0 &= \angle DVC, \\ \psi_1 &= \angle DVE, \\ \psi_2 &= \angle EVC, \end$

so that

$\psi_0 = \psi_1 + \psi_2. \qquad \qquad (1)$

Draw lines and . Angle is a central angle, but so are angles and , and
$\angle DOC = \angle DOE + \angle EOC.$

Let

$\begin \theta_0 &= \angle DOC, \\ \theta_1 &= \angle DOE, \\ \theta_2 &= \angle EOC, \end$

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 , choose three points on the circle. Draw lines and : angle is an inscribed angle. Now draw line and extend it past point so that it intersects the circle at point . Angle subtends arc on the circle.

Suppose this arc does not include point within it. Point is diametrically opposite to point . Angles 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

$\begin \psi_0 &= \angle DVC, \\ \psi_1 &= \angle EVD, \\ \psi_2 &= \angle EVC, \end$

so that

$\psi_0 = \psi_2 - \psi_1. \qquad \qquad (3)$

Draw lines and . Angle is a central angle, but so are angles and , and

$\angle DOC = \angle EOC - \angle EOD.$

Let

$\begin \theta_0 &= \angle DOC, \\ \theta_1 &= \angle EOD, \\ \theta_2 &= \angle EOC, \end$

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.)
* Ellipse
* Hyperbola
* Parabola

References

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External links

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Relationship Between Central Angle and Inscribed Angle

Munching on Inscribed Angles
at cut-the-knot

Arc Central Angle
With interactive animation


With interactive animation


With interactive animation



{{Ancient Greek mathematics

Euclidean plane geometry
Angle
Theorems about circles
Articles containing proofs