Central and inscribed angles in a circle. Demostration
The demonstration of this property can be done distinguishing several cases:
CASE 1. When the cord is a diameter, the angle is right. (Every inscribed angle that subtends a diameter is a right angle)
We use "pons asinorum".
CASE 2. When the vertex is one of the ends of the diameter and the cord draws up from the other end, the angle in center is the double of the angle in the vertex.
The previous case is not necessary but then it is simpler.
CASE 3. In the general case, depending on the disposition of the points, it is enough to add or to subtract the angles that are deduced of the previous case.
Drawing the diameter that passes through the vertex we can deduce the general case of the previous one.
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Central and inscribed angles in a circle
Central angle in a circle is twice the angle inscribed in the circle.
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