When two chords intersect inside a circle, their segment products are equal
Chord AB and chord CD intersect at point P: AP × PB = CP × PD. For secants from external point: (whole) × (external) = (whole) × (external).
Power of a Point
Power of a Point: for any point and circle, the product is constant regardless of chord direction
Power of a Point
A unified theorem for all chord, secant, and tangent relationships
Interior: two chords → segment products equal. Exterior: two secants → (whole)(external) = (whole)(external). One tangent, one secant → tangent² = (whole secant)(external secant). All follow from the same power.
Circle Vocabulary
Chord: segment with both endpoints on circle. Diameter: chord through center. Longest chord = diameter.
Circle Vocabulary
Essential terms for working with circles
Radius: center to any point on circle. Diameter: twice the radius, chord through center. Chord: any segment connecting two points on the circle. Secant: line intersecting circle at two points. Tangent: line touching circle at exactly one point (point of tangency). Arc: portion of the circle between two points.
Chord-Distance Theorems
Congruent chords are equidistant from center. Chord perpendicular from center bisects the chord.
Chord-Distance Theorems
Key relationships between chords and the center
A perpendicular from the center to a chord bisects the chord. Converse: the segment from center to the midpoint of a chord is perpendicular to the chord. Congruent chords are equidistant from the center. Chords equidistant from center are congruent.
Angles Formed by Secants and Tangents
Angles formed by two secants from external point = ½(difference of intercepted arcs)
Angles Formed by Secants and Tangents
The formula that unifies all circle angle problems
Angle formed INSIDE circle (two chords): = ½(sum of intercepted arcs). Angle formed ON circle (inscribed): = ½(intercepted arc). Angle formed OUTSIDE circle (two secants, two tangents, or secant-tangent): = ½(difference of intercepted arcs). Memory: inside=sum, outside=difference, on=one arc.
Inside circle
½ × sum of intercepted arcs
On circle (inscribed)
½ × intercepted arc
Outside circle
½ × difference of intercepted arcs
Tangent Lines from External Point
Tangent-tangent angle: two tangents from external point → angle = ½(major arc - minor arc)
Tangent Lines from External Point
Properties of two tangents drawn from the same external point
Two tangent segments from an external point are equal in length. The angle between them = ½(major arc - minor arc). The line from the external point to the center bisects the angle between the tangents. The two radii to the tangent points are perpendicular to the tangents.
Radian Measure
Radian measure: arc length = rθ. Full circle = 2π radians = 360°. 1 radian ≈ 57.3°.
Radian Measure
The natural unit of angle measurement for calculus and advanced math
1 radian: the angle subtended by an arc equal in length to the radius. 2π radians = 360° (full circle). Convert degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π. Arc length = rθ (r = radius, θ in radians). Area of sector = ½r²θ.
Circle Relationships
Concentric circles: same center, different radii. Congruent circles: same radius (may be different centers).
Circle Relationships
Two types of circle relationships — concentric and congruent
Concentric circles: share the same center but have different radii — like a bullseye. The area between two concentric circles = annulus (ring). Congruent circles: same radius, different centers — all points in one can be mapped to the other by translation. Equal radii ↔ congruent circles.
Circle Equation — General Form
Circle equation in general form: x² + y² + Dx + Ey + F = 0 → complete the square to find center and radius
Circle Equation — General Form
Converting general form to standard form by completing the square
Standard form: (x-h)² + (y-k)² = r². General form: x² + y² + Dx + Ey + F = 0. Convert: group x and y terms, complete the square for each. x² + Dx → (x + D/2)² - (D/2)². If result is positive: circle. Zero: single point. Negative: no real circle.