Circle: Key Terms and Properties

• A circle refers to a collection of points that are all the same distance from a single, fixed point in space.
• The center of the circle denotes that fixed point.
• The radius is the measurement from the center of the circle to any point on the circle.
• A circle separates the plane into an interior, an exterior, and the circle itself.

Key Terms

• A chord is a line segment connecting two points on the circle.
• A secant is a line intersecting the circle at two points.
• A tangent is a line that touches the circle at only one point.
• An arc is a continuous part of the circle.
• A sector refers to the area bounded by an arc and two radii that connect the arc's endpoints to the center.
• A segment refers to the area bounded by an arc and the chord connecting the arc's endpoints.

Angle Subtended by a Chord

• Chords of equal length in a circle create equal angles at the center of the circle.
• Chords are of equal length if they subtend equal angles at the center.
• A line drawn from the center of a circle perpendicular to a chord bisects the chord.
• A line connecting the center of a circle to the midpoint of a chord is perpendicular to the chord.
• Equal chords are an equal distance from the center of the circle.
• Chords that are the same distance from the center are equal in length.

Angle Subtended by an Arc

• An arc forms an angle at the center that is twice the angle it forms at any point on the remaining part of the circle.
• Angles are equal if they are in the same segment of a circle.
• An angle inscribed in a semicircle measures 90°.
• If a line segment connecting two points forms the same angle at two other points on the same side of the line, then all four points lie on the same circle, meaning they are concyclic.

Cyclic Quadrilaterals

• A cyclic quadrilateral is one where all four vertices lie on a circle.
• Opposite angles in in a cyclic quadrilateral add up to 180°.
• A quadrilateral where a pair of opposite angles totals 180° is a cyclic quadrilateral.

Theorems and Properties

• Only one circle can pass through three points that do not lie on the same line.
• The center of a circle that passes through three non-collinear points is found where the perpendicular bisectors of the triangle's sides intersect.
• Equal chords in congruent circles subtend equal angles at the center.
• Chords are equal if they subtend equal angles at the center of a circle.
• A line from the center of a circle that is perpendicular to a chord cuts the chord in half.
• A line drawn from the center of the circle to bisect a chord is perpendicular to it.
• Only one circle may pass through three non-collinear points.
• Chords that are the same length in a circle, or congruent circles, are equidistant from the center.
• Chords with the same distance from the center of a circle are equal in length.
• Chords are equal if corresponding arcs of a circle are congruent.
• The corresponding arcs are congruent if two chords of a circle are equal.
• Equal arcs of a circle subtend equal angles at the center.
• The angle formed by an arc at the center is double the angle formed by it at any point on the remaining part of the circle.
• Angles located in the same segment of a circle are equal.
• The angle in a semi-circle is a right angle.
• If a line segment joining two points subtends the same angle at two other points laying on the same side of the line containing the line segment, the four points lie on a circle.
• The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
• If the sum of any pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic.

Circle Terminology

• Radius: the distance from the center of the circle to any point on the circle; all radii of a circle are equal.
• Diameter: a chord that passes through the center of the circle; the diameter is twice the radius (d = 2r) and is the longest chord of the circle.
• Circumference: the distance around the circle; the formula for the circumference is C = 2πr, where r is the radius and π (pi) is approximately 3.14159.
• Arc Length: the distance along an arc of the circle.
• Central Angle: an angle formed by two radii with its vertex at the center of the circle.
• Inscribed Angle: an angle formed by two chords that share an endpoint on the circle; its vertex lies on the circle.

Relationships and Theorems

• The measure of an inscribed angle is half the measure of its intercepted central angle.
• A tangent to a circle if perpendicular to the radius at the point of tangency.
• When two tangents are drawn to a circle from the same external point, the lengths of the tangent segments from the external point to the points of tangency are equal.
• The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

Circle Equations

• A circle with its center at the origin (0, 0) and a radius r has the equation x² + y² = r².
• A circle with its center at (h, k) and a radius r has the equation (x - h)² + (y - k)² = r².

Concyclic Points

• Points are referred to as concyclic when they are located on the same circle.
• Four points are concyclic if the angles created by a line segment connecting any two of the points are equal at the other two points, and all points are on the same side of the line segment.

Common Tangents

• A direct common tangent is a line that touches two circles without crossing the line segment that connects their centers.
• A transverse common tangent is a line that touches two circles and crosses the line segment connecting their centers.
• Common tangent lengths can be determined through geometric relationships and the Pythagorean theorem.