**Important Concepts and Formulas**

1. A circle is the collection of all
points in a plane, which are equidistant from a fixed point in the plane. The fixed
point is called the center of the circle.

2. Equal chords of a circle (or of
congruent circles) subtend equal angles at the center of the circle.

3. If the angles subtended by two chords
of a circle (or of congruent circles) at the center are equal, the chords are
equal.

4. The perpendicular from the center of
a circle to a chord bisects the chords.

5. The line drawn from the center of a
circle to bisect a chord is perpendicular to the chord.

6. There is one and only one circle
passing through three non-collinear points.

7. Equal chords of a circle (or of
congruent circles) are equidistant from the center of the circle.

8. Chords equidistant from the center (or
corresponding centers) of a circle (or of congruent circles) are equal.

9. If two arcs of a circle are
congruent, then their corresponding chords are equal and conversely if the
chords of a circle are equal, then their corresponding arcs (minor and major)
are congruent.

10. Congruent arcs of a circle subtend
equal angles at the center.

11. The angle subtended by an arc at the
center is double the angle subtended by it at any point on the remaining part
of the circle.

12. Angles in the same segment of a
circle are equal.

13. Angle in a semicircle is a right
angle.

14. If a line segment joining two points
subtends equal angles at two other points lying on the same side of the line containing
the line segment, the four points lie on a circle.

15. The sum of a pair of opposite angles
of a cyclic quadrilateral is 180°.

16. If sum of a pair of opposite angles
of a quadrilateral is 180°, the quadrilateral is a cyclic
quadrilateral.

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