Theorem The angle inscribed in a circle is equal to half the corresponding central angle.
Evidence. Let there be a circle centered at the point O and an angle ABC inscribed in this circle, so that one of the sides of the angle passes through the center of the circle.
Connect point A with the center of the circle with point O. Δ ABO is isosceles (BO = AO as radii). Therefore, ∠OBA = ∠OAB. The external angle at the vertex O, the angle AOC is equal to the sum of the angles OBA and OAB. Means
![Angles inscribed in a circle. Property](/th/25/blogs/id3416/1_ygolvpisansvojstv2.jpg)
In general, there may be two options where the sides of the corner do not pass through the center of the circle. We will hold the auxiliary diameter BD
Option 1:
![Angles inscribed in a circle. Property](/th/25/blogs/id3416/2_ygolvpisansvojstv3.jpg)
Then
![Angles inscribed in a circle. Property](/th/25/blogs/id3416/3_ygolvpisansvojstv4.jpg)
Option 2:
![Angles inscribed in a circle. Property](/th/25/blogs/id3416/4_ygolvpisansvojstv5.jpg)
Then
![Angles inscribed in a circle. Property](/th/25/blogs/id3416/5_ygolvpisansvojstv6.jpg)
The theorem is proved.
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Planometry
Terms: Planometry