Circle theorams

Theorem 1: The angle subtended by an arc at the center of a circle is double the angle subtended at any point on the circumference.

Theorem 2: Angles in the same segment of a circle are equal, which implies that if two angles are subtended by the same arc, then they will be congruent.

Theorem 3: The opposite angles of a cyclic quadrilateral are supplementary, meaning that their measures add up to 180 degrees.

Theorem 4: The external angle of a triangle formed by two chords intersecting outside the circle is equal to the sum of the internal opposite angles.

Theorem 5: The tangent to a circle at any point is perpendicular to the radius drawn to that point, which establishes the foundational relationship between the radius and tangent line.

Theorem 6: The angle subtended by an arc at the center of the circle is twice the angle subtended at any point on the remaining part of the circle, providing insight into the relationship between central and inscribed angles.

Theorem 7: The lengths of two chords from a point inside the circle to the endpoints of the chords are related, specifically, the products of the lengths of the segments of each chord are equal to one another.

Theorem 8: The angle formed between a tangent and a chord through the point of contact is equal to the angle in the alternate segment of the circle, illustrating the connection between tangents and secants.

Theorem 9: The perpendicular from the center of the circle to a chord bisects the chord, demonstrating how centrality affects the symmetry of the chord and its segments.

Theorem 10: The area of a sector of a circle is proportional to the angle subtended at the center, providing a valuable formula for calculating areas of circular segments based on their angular measures.

Theorem 11: The lengths of two tangents drawn from an external point to a circle are equal, establishing the principle that distances from external points to a circle maintain equality when intersecting tangents.