Inscribed angle

The inscribed angle Template:Mvar circle. Template:Legend-line Template:Legend-line Template:Legend-line

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 (but of an angle of a triangle not inscribed in a circle).

Theorem

Statement

For fixed points Template:Mvar and Template:Mvar, the set of points M in the plane, for which the angle Template:Math is equal to α, is an arc of a circle. The measure of Template:Math, where Template:Mvar is the center of the circle, is Template:Math.

The inscribed angle theorem states that an angle Template:Mvar inscribed in a circle is half of the central angle Template:Math 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

Case: One chord is a diameter

Let Template:Mvar be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them Template:Mvar and Template:Mvar. Designate point Template:Mvar to be diametrically opposite point Template:Mvar. Draw chord Template:Mvar, a diameter containing point Template:Mvar. Draw chord Template:Mvar. Angle Template:Math is an inscribed angle that intercepts arc Template:Mvar; denote it as Template:Mvar. Draw line Template:Mvar. Angle Template:Math is a central angle that also intercepts arc Template:Mvar; denote it as Template:Mvar.

Lines Template:Mvar and Template:Mvar are both radii of the circle, so they have equal lengths. Therefore, triangle Template:Math is isosceles, so angle Template:Math and angle Template:Math are equal.

Angles Template:Math and Template:Math are supplementary, summing to a straight angle (180°), so angle Template:Math measures Template:Math.

The three angles of triangle Template:Math [[sum of angles of a triangle|must sum to Template:Math]]:

$$(180^{\circ }-\theta )+\psi +\psi =180^{\circ }.$$

Adding $\theta -180^{\circ }$ to both sides yields

$$2\psi =\theta .$$

Inscribed angles with the center of the circle in their interior

Case: Center interior to angle Template:Legend-line Template:Legend-line Template:Legend-line

Given a circle whose center is point Template:Mvar, choose three points Template:Mvar on the circle. Draw lines Template:Mvar and Template:Mvar: angle Template:Math is an inscribed angle. Now draw line Template:Mvar and extend it past point Template:Mvar so that it intersects the circle at point Template:Mvar. Angle Template:Math intercepts arc Template:Mvar on the circle.

Suppose this arc includes point Template:Mvar within it. Point Template:Mvar is diametrically opposite to point Template:Mvar. Angles Template:Math 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,

$$\mathrm{\angle }DVC=\mathrm{\angle }DVE+\mathrm{\angle }EVC.$$

then let

$$\begin{array}{cc}{\psi }_0& =\mathrm{\angle }DVC,\\ {\psi }_1& =\mathrm{\angle }DVE,\\ {\psi }_2& =\mathrm{\angle }EVC,\end{array}$$

so that

$${\psi }_0={\psi }_1+{\psi }_2.(1)$$

Draw lines Template:Mvar and Template:Mvar. Angle Template:Math is a central angle, but so are angles Template:Math and Template:Math, and $$\mathrm{\angle }DOC=\mathrm{\angle }DOE+\mathrm{\angle }EOC.$$

Let

$$\begin{array}{cc}{\theta }_0& =\mathrm{\angle }DOC,\\ {\theta }_1& =\mathrm{\angle }DOE,\\ {\theta }_2& =\mathrm{\angle }EOC,\end{array}$$

so that

$${\theta }_0={\theta }_1+{\theta }_2.(2)$$

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

$${\theta }_0=2{\psi }_1+2{\psi }_2=2({\psi }_1+{\psi }_2)$$

therefore, by equation (1),

$${\theta }_0=2{\psi }_0.$$

Inscribed angles with the center of the circle in their exterior

Case: Center exterior to angle Template:Legend-line Template:Legend-line Template:Legend-line

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 Template:Mvar, choose three points Template:Mvar on the circle. Draw lines Template:Mvar and Template:Mvar: angle Template:Math is an inscribed angle. Now draw line Template:Mvar and extend it past point Template:Mvar so that it intersects the circle at point Template:Mvar. Angle Template:Math intercepts arc Template:Mvar on the circle.

Suppose this arc does not include point Template:Mvar within it. Point Template:Mvar is diametrically opposite to point Template:Mvar. Angles Template:Math 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,

$$\mathrm{\angle }DVC=\mathrm{\angle }EVC-\mathrm{\angle }EVD.$$

then let

$$\begin{array}{cc}{\psi }_0& =\mathrm{\angle }DVC,\\ {\psi }_1& =\mathrm{\angle }EVD,\\ {\psi }_2& =\mathrm{\angle }EVC,\end{array}$$

so that

$${\psi }_0={\psi }_2-{\psi }_1.(3)$$

Draw lines Template:Mvar and Template:Mvar. Angle Template:Math is a central angle, but so are angles Template:Math and Template:Math, and

$$\mathrm{\angle }DOC=\mathrm{\angle }EOC-\mathrm{\angle }EOD.$$

Let

$$\begin{array}{cc}{\theta }_0& =\mathrm{\angle }DOC,\\ {\theta }_1& =\mathrm{\angle }EOD,\\ {\theta }_2& =\mathrm{\angle }EOC,\end{array}$$

so that

$${\theta }_0={\theta }_2-{\theta }_1.(4)$$

From Part One we know that ${\theta }_1=2{\psi }_1$ and that ${\theta }_2=2{\psi }_2$. Combining these results with equation (4) yields $${\theta }_0=2{\psi }_2-2{\psi }_1$$ therefore, by equation (3), $${\theta }_0=2{\psi }_0.$$

Animated gif of proof of the inscribed angle theorem. The large triangle that is inscribed in the circle gets subdivided into three smaller triangles, all of which are isosceles because their upper two sides are radii of the circle. Inside each isosceles triangle the pair of base angles are equal to each other, and are half of 180° minus the apex angle at the circle's center. Adding up these isosceles base angles yields the theorem, namely that the inscribed angle, Template:Mvar, is half the central angle, Template:Mvar.

Corollary

The angle Template:Mvar 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.

Applications

Proof without words 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 for ellipses, hyperbolas and parabolas

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

References

• Template:Cite book
• Template:Cite book
• Template:Cite book

External links

• Template: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.github.io