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Classical Geometry – Circle
Definition: Circle is defined as a set of points with the same distance to a fixed point O on a plane.
Let E be a plane and O be a fixed point on E. If C is a set such that C = { p | the distance from point p to point O is r. p E } where r is a constant. Then C is a circle.
Definition: ( Arc )
Given 2 points A and B on a circle, arc AB usually means the collection of points from A to B on the circle ( for the shorter route on the circle shown as below ). And it is denoted as . This notation is also used to represent the length of the arc.
If we want to represent the longer route from A to B on the circle, usually we use one more point on the circle to denote that arc, for example .
Please note that is different from . They stand for different arcs. It is necessary to follow the consecutive order along the circle when you want to specify an arc.
Definition: ( Central angle ) Given 2 points A and B on the circle, AOB is a central angle corresponding to . And the magnitude of a central angle AOB is defined by
where r is the radius of the circle.
And some ancient people divided equally the circle into 360 pieces. The central angle corresponding to each piece is denoted as 1o . Please check how to convert those metric systems.
Definition: ( Inscribed angle ) As shown below, point A, B, and C are on the circle. BCA is one of the inscribed angles corresponding to .
Theorem: Given , AOB = 2 ACB where AOB is the corresponding central angle and ACB is its inscribed angle.
Proof:
It is sufficient to prove the following two cases for AOB = 2 BCA:
1. Consider the following diagram by drawing the line :
(1) Look at . We have = = r , OCB + OBC = BOD where r is the radius of the circle .
Then BOD = 2 OCB (2) Similarly, look at : = , OCA + OAC = AOD Then AOD = 2 OAC
(3) AOB = AOD - AOB = 2 ( OAC - OCB ) = 2 BCA
2. Consider the following diagram by drawing the line :
(1) In , = BOD = 2 BCO (2) Similarly for , = AOD = 2 ACO (3) BOA = BOD + AOD = 2 ( BCO + ACO ) = 2 BCA
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