Inscribed angle#Theorem

File:ArcCapable.gif

The inscribed angle theorem states that an angle {{mvar|θ}} inscribed in a circle is half of the central angle {{math|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.

File:InscribedAngle 1ChordDiam.svg

Let {{mvar|O}} be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them {{mvar|V}} and {{mvar|A}}. Designate point {{mvar|B}} to be diametrically opposite point {{mvar|V}}. Draw chord {{Mvar|VB}}, a diameter containing point {{Mvar|O}}. Draw chord {{Mvar|VA}}. Angle {{Math|∠BVA}} is an inscribed angle that intercepts arc {{Mvar|{{overarc|AB}}}}; denote it as {{Mvar|ψ}}. Draw line {{mvar|OA}}. Angle {{math|∠BOA}} is a central angle that also intercepts arc {{Mvar|{{overarc|AB}}}}; denote it as {{mvar|θ}}.

Lines {{mvar|OV}} and {{mvar|OA}} are both radii of the circle, so they have equal lengths. Therefore, triangle {{math|△VOA}} is isosceles, so angle {{math|∠BVA}} and angle {{math|∠VAO}} are equal.

Angles {{math|∠BOA}} and {{math|∠AOV}} are supplementary, summing to a straight angle (180°), so angle {{math|∠AOV}} measures {{math|180° − θ}}.

The three angles of triangle {{math|△VOA}} sum of angles of a triangle:

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

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

$$2\backslash psi\; =\; \backslash theta.$$

[[File:Circle-angles-21add-inscribed.svg|thumb|class=skin-invert-image|

Case: Center interior to angle

{{legend-line|solid blue|{{math|1=ψ{{sub|0}} = ∠DVC, θ{{sub|0}} = ∠DOC}}}}

{{legend-line|solid green|{{math|1=ψ{{sub|1}} = ∠EVD, θ{{sub|1}} = ∠EOD}}}}

{{legend-line|solid red|{{math|1=ψ{{sub|2}} = ∠EVC, θ{{sub|2}} = ∠EOC}}}}

]]

Given a circle whose center is point {{mvar|O}}, choose three points {{mvar|V, C, D}} on the circle. Draw lines {{mvar|VC}} and {{mvar|VD}}: angle {{math|∠DVC}} is an inscribed angle. Now draw line {{mvar|OV}} and extend it past point {{mvar|O}} so that it intersects the circle at point {{mvar|E}}. Angle {{math|∠DVC}} intercepts arc {{mvar|{{overarc|DC}}}} on the circle.

Suppose this arc includes point {{mvar|E}} within it. Point {{mvar|E}} is diametrically opposite to point {{mvar|V}}. Angles {{math|∠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,

$$\backslash angle\; DVC\; =\; \backslash angle\; DVE\; +\; \backslash angle\; EVC.$$

then let

$$\backslash begin\{align\}$$

\psi_0 &= \angle DVC, \\

\psi_1 &= \angle DVE, \\

\psi_2 &= \angle EVC,

\end{align}

so that

$$\backslash psi\_0\; =\; \backslash psi\_1\; +\; \backslash psi\_2.\; \backslash qquad\; \backslash qquad\; (1)$$

Draw lines {{mvar|OC}} and {{mvar|OD}}. Angle {{math|∠DOC}} is a central angle, but so are angles {{math|∠DOE}} and {{math|∠EOC}}, and

$$\backslash angle\; DOC\; =\; \backslash angle\; DOE\; +\; \backslash angle\; EOC.$$

Let

$$\backslash begin\{align\}$$

\theta_0 &= \angle DOC, \\

\theta_1 &= \angle DOE, \\

\theta_2 &= \angle EOC,

\end{align}

so that

$$\backslash theta\_0\; =\; \backslash theta\_1\; +\; \backslash theta\_2.\; \backslash qquad\; \backslash qquad\; (2)$$

From Part One we know that $\backslash theta\_1\; =\; 2\; \backslash psi\_1$ and that $\backslash theta\_2\; =\; 2\; \backslash psi\_2$. Combining these results with equation (2) yields

$$\backslash theta\_0\; =\; 2\; \backslash psi\_1\; +\; 2\; \backslash psi\_2\; =\; 2(\backslash psi\_1\; +\; \backslash psi\_2)$$

therefore, by equation (1),

$$\backslash theta\_0\; =\; 2\; \backslash psi\_0.$$

[[Image:InscribedAngle CenterCircleExtV2.svg|thumb|Case: Center exterior to angle

{{legend-line|solid orange|{{math|1=ψ{{sub|0}} = ∠DVC, θ{{sub|0}} = ∠DOC}}}}

{{legend-line|solid green|{{math|1=ψ{{sub|1}} = ∠EVD, θ{{sub|1}} = ∠EOD}}}}

{{legend-line|solid blue|{{math|1=ψ{{sub|2}} = ∠EVC, θ{{sub|2}} = ∠EOC}}}}

]]

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 {{mvar|O}}, choose three points {{mvar|V, C, D}} on the circle. Draw lines {{mvar|VC}} and {{mvar|VD}}: angle {{math|∠DVC}} is an inscribed angle. Now draw line {{mvar|OV}} and extend it past point {{mvar|O}} so that it intersects the circle at point {{mvar|E}}. Angle {{math|∠DVC}} intercepts arc {{mvar|{{overarc|DC}}}} on the circle.

Suppose this arc does not include point {{mvar|E}} within it. Point {{mvar|E}} is diametrically opposite to point {{mvar|V}}. Angles {{math|∠EVD, ∠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,

$$\backslash angle\; DVC\; =\; \backslash angle\; EVC\; -\; \backslash angle\; EVD.$$

then let

$$\backslash begin\{align\}$$

\psi_0 &= \angle DVC, \\

\psi_1 &= \angle EVD, \\

\psi_2 &= \angle EVC,

\end{align}

so that

$$\backslash psi\_0\; =\; \backslash psi\_2\; -\; \backslash psi\_1.\; \backslash qquad\; \backslash qquad\; (3)$$

Draw lines {{mvar|OC}} and {{mvar|OD}}. Angle {{math|∠DOC}} is a central angle, but so are angles {{math|∠EOD}} and {{math|∠EOC}}, and

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

Let

$$\backslash begin\{align\}$$

\theta_0 &= \angle DOC, \\

\theta_1 &= \angle EOD, \\

\theta_2 &= \angle EOC,

\end{align}

so that

$$\backslash theta\_0\; =\; \backslash theta\_2\; -\; \backslash theta\_1.\; \backslash qquad\; \backslash qquad\; (4)$$

From Part One we know that $\backslash theta\_1\; =\; 2\; \backslash psi\_1$ and that $\backslash theta\_2\; =\; 2\; \backslash psi\_2$. Combining these results with equation (4) yields

$$\backslash theta\_0\; =\; 2\; \backslash psi\_2\; -\; 2\; \backslash psi\_1$$

therefore, by equation (3),

$$\backslash theta\_0\; =\; 2\; \backslash psi\_0.$$

File:Animated gif of proof of the inscribed angle theorem.gif

Image:Inscribed angle theorem4.svg

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.

File:cyclic_quadrilateral_supplementary_angles_visual_proof.svg using the inscribed angle theorem that opposite angles of a cyclic quadrilateral are supplementary:
2𝜃 + 2𝜙 = 360° ∴ 𝜃 + 𝜙 = 180°]]

The inscribed angle theorem is used in many proofs of elementary Euclidean geometry of the plane. A special case of the theorem is Thales's 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 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

• {{Cite book | author-link = C. Stanley Ogilvy | last = Ogilvy | first = C. S. | year = 1990 | title = Excursions in Geometry | publisher = Dover | isbn = 0-486-26530-7 | pages = 17–23 }}
• {{cite book |vauthors=Gellert W, Küstner H, Hellwich M, Kästner H | title = The VNR Concise Encyclopedia of Mathematics | publisher = Van Nostrand Reinhold | location = New York | isbn = 0-442-22646-2 | pages = 172 | year = 1977}}
• {{cite book |first=Edwin E. |last=Moise |author-link=Edwin E. Moise |title=Elementary Geometry from an Advanced Standpoint |location=Reading |publisher=Addison-Wesley |edition=2nd |year=1974 |isbn=0-201-04793-4 |pages=192–197 }}

• {{MathWorld |urlname=InscribedAngle |title=Inscribed Angle}}
• [http://www.mathalino.com/reviewer/plane-geometry/relationship-between-central-angle-and-inscribed-angle Relationship Between Central Angle and Inscribed Angle]
• [http://www.cut-the-knot.org/pythagoras/Munching/inscribed.shtml Munching on Inscribed Angles] at cut-the-knot
• [https://web.archive.org/web/20061030174939/http://www.mathopenref.com/arccentralangle.html Arc Central Angle] With interactive animation
• [http://www.mathopenref.com/arcperipheralangle.html Arc Peripheral (inscribed) Angle] With interactive animation
• [http://www.mathopenref.com/arccentralangletheorem.html Arc Central Angle Theorem] With interactive animation
• [https://bookofproofs.github.io/branches/geometry/elements-euclid/book--3-circles/inscribed-angle-theorem.html At bookofproofs.github.io]

{{Ancient Greek mathematics}}

Category:Euclidean plane geometry

Category:Angle

Category:Theorems about circles

Category:Articles containing proofs