Английская Википедия: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 subtending the same arc.
The inscribed angle theorem appears as Proposition 20 on Book 3 of Euclid's Elements.
Theorem
Statement
The inscribed angle theorem states that an angle Шаблон:Mvar inscribed in a circle is half of the central angle Шаблон:Math that subtends the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the circle.
Proof
Inscribed angles where one chord is a diameter
Let Шаблон:Mvar be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them Шаблон:Mvar and Шаблон:Mvar. Draw line Шаблон:Mvar and extended past Шаблон:Mvar so that it intersects the circle at point Шаблон:Mvar which is diametrically opposite the point Шаблон:Mvar. Draw an angle whose vertex is point Шаблон:Mvar and whose sides pass through points Шаблон:Mvar.
Draw line Шаблон:Mvar. Angle Шаблон:Math is a central angle; call it Шаблон:Mvar. Lines Шаблон:Mvar and Шаблон:Mvar are both radii of the circle, so they have equal lengths. Therefore, triangle Шаблон:Math is isosceles, so angle Шаблон:Math (the inscribed angle) and angle Шаблон:Math are equal; let each of them be denoted as Шаблон:Mvar.
Angles Шаблон:Math and Шаблон:Math are supplementary, summing to a straight angle (180°), so angle Шаблон:Math measures Шаблон:Math.
The three angles of triangle Шаблон:Math must sum to Шаблон:Math:
<math display=block>(180^\circ - \theta) + \psi + \psi = 180^\circ.</math>
Adding <math>\theta - 180^\circ</math> to both sides yields
<math display=block>2\psi = \theta.</math>
Inscribed angles with the center of the circle in their interior
Given a circle whose center is point Шаблон:Mvar, choose three points Шаблон:Mvar on the circle. Draw lines Шаблон:Mvar and Шаблон:Mvar: angle Шаблон:Math is an inscribed angle. Now draw line Шаблон:Mvar and extend it past point Шаблон:Mvar so that it intersects the circle at point Шаблон:Mvar. Angle Шаблон:Math subtends arc Шаблон:Mvar on the circle.
Suppose this arc includes point Шаблон:Mvar within it. Point Шаблон:Mvar is diametrically opposite to point Шаблон:Mvar. Angles Шаблон: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,
<math display=block> \angle DVC = \angle DVE + \angle EVC. </math>
then let
<math display=block>\begin{align}
\psi_0 &= \angle DVC, \\ \psi_1 &= \angle DVE, \\ \psi_2 &= \angle EVC,
\end{align}</math>
so that
<math display=block> \psi_0 = \psi_1 + \psi_2. \qquad \qquad (1) </math>
Draw lines Шаблон:Mvar and Шаблон:Mvar. Angle Шаблон:Math is a central angle, but so are angles Шаблон:Math and Шаблон:Math, and <math display=block> \angle DOC = \angle DOE + \angle EOC. </math>
Let
<math display=block>\begin{align}
\theta_0 &= \angle DOC, \\ \theta_1 &= \angle DOE, \\ \theta_2 &= \angle EOC,
\end{align}</math>
so that
<math display=block> \theta_0 = \theta_1 + \theta_2. \qquad \qquad (2) </math>
From Part One we know that <math> \theta_1 = 2 \psi_1 </math> and that <math> \theta_2 = 2 \psi_2 </math>. Combining these results with equation (2) yields
<math display=block> \theta_0 = 2 \psi_1 + 2 \psi_2 = 2(\psi_1 + \psi_2) </math>
therefore, by equation (1),
<math display=block> \theta_0 = 2 \psi_0. </math>
Inscribed angles with the center of the circle in their exterior
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 Шаблон:Mvar, choose three points Шаблон:Mvar on the circle. Draw lines Шаблон:Mvar and Шаблон:Mvar: angle Шаблон:Math is an inscribed angle. Now draw line Шаблон:Mvar and extend it past point Шаблон:Mvar so that it intersects the circle at point Шаблон:Mvar. Angle Шаблон:Math subtends arc Шаблон:Mvar on the circle.
Suppose this arc does not include point Шаблон:Mvar within it. Point Шаблон:Mvar is diametrically opposite to point Шаблон:Mvar. Angles Шаблон: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,
<math display=block> \angle DVC = \angle EVC - \angle EVD </math>.
then let
<math display=block>\begin{align}
\psi_0 &= \angle DVC, \\ \psi_1 &= \angle EVD, \\ \psi_2 &= \angle EVC,
\end{align}</math>
so that
<math display=block> \psi_0 = \psi_2 - \psi_1. \qquad \qquad (3) </math>
Draw lines Шаблон:Mvar and Шаблон:Mvar. Angle Шаблон:Math is a central angle, but so are angles Шаблон:Math and Шаблон:Math, and
<math display=block> \angle DOC = \angle EOC - \angle EOD. </math>
Let
<math display=block>\begin{align}
\theta_0 &= \angle DOC, \\ \theta_1 &= \angle EOD, \\ \theta_2 &= \angle EOC,
\end{align}</math>
so that
<math display=block> \theta_0 = \theta_2 - \theta_1. \qquad \qquad (4) </math>
From Part One we know that <math> \theta_1 = 2 \psi_1 </math> and that <math> \theta_2 = 2 \psi_2 </math>. Combining these results with equation (4) yields <math display=block> \theta_0 = 2 \psi_2 - 2 \psi_1 </math> therefore, by equation (3), <math display=block> \theta_0 = 2 \psi_0. </math>
Corollary
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
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' 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.)
References
External links
- Шаблон: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
Шаблон:Ancient Greek mathematics
