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 intercepting 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.
Theorem
Statement
The inscribed angle theorem states that an angle inscribed in a circle is half of the central angle that intercepts the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the same arc of 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. Designate point to be diametrically opposite point. Draw chord, a diameter containing point. Draw chord. Angle is an inscribed angle that intercepts arc ; denote it as. Draw line. Angle is a central angle that also intercepts arc ; denote it as.Lines and are both radii of the circle, so they have equal lengths. Therefore, triangle is isosceles, so angle and angle are equal.
Angles and are supplementary, summing to a straight angle, so angle measures.
The three angles of triangle must sum to :
Adding to both sides yields
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 intercepts 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,
then let
so that
Draw lines and. Angle is a central angle, but so are angles and, and
Let
so that
From Part One we know that and that. Combining these results with equation yields
therefore, by equation,
Inscribed angles with the center of the circle in their exterior
[Image:InscribedAngle CenterCircleExtV2.svg|thumb|Case: Center exterior to angle]
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 intercepts 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,
then let
so that
Draw lines and. Angle is a central angle, but so are angles and, and
Let
so that
From Part One we know that and that. Combining these results with equation yields
therefore, by equation,
Corollary
[Image:Inscribed angle theorem4.svg|thumb|The angle between a chord and a tangent is half the arc belonging to the chord.]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.