Inscribed angle theorem

Inscribed angle theorem

In geometry, 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 apex 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 any quadrilateral inscribed in a circle must 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

The proof of the inscribed angle theorem is relatively simple. 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 isoceles triangle, made from two radii of the circle and the second leg of the inscribed angle. Since two angles in an isoceles 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 arbitrary inscribed angles by drawing a diameter from its apex. This converts the general problem into two sub-cases in which a diameter is a leg. Adding the two subangles again yields the result that the inscribed angle is half of the central angle.

Corollaries

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.

Bibliography

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