# Inscribed angle

In geometry, an **inscribed angle** is the angle formed in the interior of a circle when two secant lines 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 on Book 3 of Euclid’s "Elements".

## Theorem[edit]

### Statement[edit]

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[edit]

#### Inscribed angles where one chord is a diameter[edit]

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 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,

Subtract 180° from both sides,

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]

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,

then let

so that

Draw lines *OC* and *OD*. Angle *DOC* is a central angle, but so are angles *DOE* and *EOC*, and

Let

so that

From Part One we know that and that . Combining these results with equation (2) yields

therefore, by equation (1),

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

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,

- .

then let

so that

Draw lines *OC* and *OD*. Angle *DOC* is a central angle, but so are angles *EOD* and *EOC*, and

Let

so that

From Part One we know that and that . Combining these results with equation (4) yields

therefore, by equation (3),

### Corollary[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.

## Applications[edit]

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[edit]

Inscribed angle theorems exist for ellipses, hyperbolas and parabolas, too. The essential differences are the measurements of an angle. (An angle is considered as a pair of intersecting lines.)

## 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. - 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[edit]

- 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.org