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. The inscribed angle theorem, sometimes called the central angle theorem, relates the measure of an inscribed angle to that of the central angle intercepting the same arc. Its oldest appearance is in Propositions 20–21 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 θ inscribed in a circle is half of the central angle 2θ 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 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. Designate point B to be diametrically opposite point V. Draw chord VB, a diameter containing point O. Draw chord VA. Angle ∠BVA is an inscribed angle that intercepts arc AB; denote it as ψ. Draw line OA. Angle ∠BOA is a central angle that also intercepts arc AB; denote it as θ. Lines OV and OA are both radii of the circle, so they have equal lengths. Therefore, triangle △VOA is isosceles, so angle ∠BVA and angle ∠VAO are equal. 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 ∘ − θ ) + ψ + ψ = 180 ∘ . {\displaystyle (180^{\circ }-\theta )+\psi +\psi =180^{\circ }.}
Adding θ − 180 ∘ {\displaystyle \theta -180^{\circ }} to both sides yields
2 ψ = θ . {\displaystyle 2\psi =\theta .}
Inscribed angles with the center of the circle in their interior
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 intercepts 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,
∠ D V C = ∠ D V E + ∠ E V C . {\displaystyle \angle DVC=\angle DVE+\angle EVC.}
then let
ψ 0 = ∠ D V C , ψ 1 = ∠ D V E , ψ 2 = ∠ E V C , {\displaystyle {\begin{aligned}\psi _{0}&=\angle DVC,\\\psi _{1}&=\angle DVE,\\\psi _{2}&=\angle EVC,\end{aligned}}}
so that
ψ 0 = ψ 1 + ψ 2 . ( 1 ) {\displaystyle \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
∠ D O C = ∠ D O E + ∠ E O C . {\displaystyle \angle DOC=\angle DOE+\angle EOC.}
Let
θ 0 = ∠ D O C , θ 1 = ∠ D O E , θ 2 = ∠ E O C , {\displaystyle {\begin{aligned}\theta _{0}&=\angle DOC,\\\theta _{1}&=\angle DOE,\\\theta _{2}&=\angle EOC,\end{aligned}}}
so that
… excerpt ends here. Continue reading the full article.
