Inscribed angle theorem

Builds on Circle Oriented angle Angle at a point

Statement

Let $A$ , $B$ and $V$ be distinct points of a circle with the centre $O$ in the Euclidean plane $\mathbb{R}^2$ . Let $\measuredangle$ be the oriented angle, with values in $\mathbb{R} / 2\pi\mathbb{Z}$ . Then $\measuredangle AOB = 2 \, \measuredangle AVB.$

Remarks

For the unoriented angles, with values in $[0, \pi]$ : if $V$ and $O$ are not on opposite sides of the line $AB$ , then $\angle AVB = \frac{1}{2} \angle AOB$ . If they are on opposite sides, then $\angle AVB = \pi - \frac{1}{2} \angle AOB$ . Thus the inscribed angle does not change when $V$ moves on one arc from $A$ to $B$ . Thales's theorem is the case where $AB$ is a diameter. The oldest known version is in Propositions 20 and 21 of Book III of Euclid's Elements.

Reading path

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Angle at a point concept
Circle concept
Oriented angle concept
Inscribed angle theorem statement

Where this leads

1 page builds on this one.

Angles in the same segment are equal

Tag	`0GA1V8NN8C`
URI	`m2://0GA1V8NN8C`
Aliases	`central-angle-theorem`
Kind	statement, theorem
Writing	W0 stub
MSC	51M04
Status	This copy of the site has no build data. Run `m2 status 0GA1V8NN8C`.
Uses	Circle, Oriented angle, Angle at a point

Inscribed angle theorem

Statement

Remarks

Reading path

Where this leads

Further reading

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