Final answer:
The angle subtended by a chord at the center of a circle (central angle) is twice the angle subtended at the circumference (inscribed angle) because in the created isosceles triangle from the center, the central angle is twice one of the equal angles at the base by the properties of isosceles triangles and the sum of triangle angles.
Step-by-step explanation:
The statement that the angle subtended by a chord at the center of a circle is twice the angle subtended at the circumference is a fundamental theorem in circle geometry. To prove this, let us consider a circle with center O and a chord AB. When we draw radii, OA and OB, we create an isosceles triangle OAB because OA and OB are both radii of the circle and thus have the same length. If we then take any point C on the circumference of the circle that is not on the chord AB, and draw lines AC and BC, we have another triangle, ABC.
In triangle OAB, the angle at the center ∠AOB is subtended by the arc ACB. In triangle ABC, the angle at the circumference ∠ACB is subtended by the same arc ACB. Since OAB is an isosceles triangle, the angles at O, ∠OAB and ∠OBA, are equal. By the properties of isosceles triangles and the fact that the sum of angles in a triangle is 180 degrees, we can say that the angle ∠AOB is twice the angle of ∠ACB.
As a result, we can deduce that the central angle is twice the inscribed angle that subtends the same arc, adhering to the principles of circle geometry. This conclusion can be generalized to state that for any chord and any point on the circumference of the circle, the central angle will always be double the angle at the circumference.