Circles

• A chord is a segment with endpoints on a circle
• A diameter is a chord that has a center of a circle
• A tangent is a line in the plane of a circle that intersects at one point
• A point of tangency is where a tangent intersects
• Coplanar circles at 1 point of intersection are tangent circles
• Coplanar circles with no points of intersection are concentric
• Common tangents are lines, rays, or segments that are tangent to 2 coplanar circles
• Internal common tangents intersects segments joining the center of 2 circles; they are in between usually
• External common tangents do not intersect the segment joining the center of 2 circles
• Radius-Tangent Theorem is a line is tangent to a circle if the line is perpendicular to the radius of a circle at its endpoint
• Radius-Tangent Theorem Converse is if a line is perpendicular to a a radius on a circle at a point on a circle, its tangent
• Tangent Segments Theorem is if 2 tangents are drawn to a circle form the same point outside, the segments are congruent
• Arc Addition Postulate is that the measure of an arc formed by 2 adjacent arcs is the sum of measures of 2 arcs
• Two circles that are congruent have the same radius; two arcs are congruent if they have the same measures and are arcs on the same circle, or two congruent circles
• The Perpendicular Bisector of a Chord Theorem Converse is if a chord is a perpendicular bisector, the first chord is the diameter
• Two chords are congruent if they are equidistant from the center
• An inscribed angle is an angle whose vertex is on a circle; has sides containing chords of circle
• An intercepted arc is an arc that lies in the interior of an inscribed angle; endpoints are on the angle
• Inscribed polygon is a polygon with all of its vertices on a circle
• Circumscribed circle is a circle containing vertices of the inscribed polygon
• Only certain quadrilaterals can be inscribed in a circle
• The Inscribed Right Triangle Theorem is if a right triangle is inscribed in a circle, the hypotenuse is the diameter
• The Inscribed Right Triangle Theorem Converse is if one side of an inscribed triangle is the diameter of a circle, the triangle is a right triangle
• The Inscribed Quadrilateral Theorem is a quadrilateral can be inscribed in a circle if its opposite angles are supplementary
• Two intercepted lines can intersect on the circle, or inside/outside a circle
• You can write the standard equation if you know its radius and the coordinates of its center.
• You can use the Distance Formula to find the distance between the center and a point on a circle.
• The standard equation of a circle with a center and radius is (x-h)² + (y-k)² = r²
• A segment of a chord is a segment of two chords that intersect
• A secant segment is a segment that contains a chord of a circle, with one endpoint outside the circle.
• The Segments of Secants Theorem says if two secant segments share the same endpoint outside a circle, the product of the lengths of one secant segment and its external segment equals the products of the lengths of the other secant segment and its external segment

  
• Tangent and Chord Intersection Theorem is if a tangent and chord intersect at a point on a circle, the measure of each angle formed is half the measure of its intercepted arc

  
• Angles Inside the Circle Theorem is if two chords intersect inside a circle, the measure of each angle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle

  
• The Angles Outside the Circle Theorem is if a tangent and a secant OR two tangents OR two secants intersect outside a circle, then the measure of the angle formed is half the difference of the measures of the intercepted arcs

  
• Inscribed Angle Theorem says that the measure of an inscribed angle is half the measure of its intercepted arc

  
• Two inscribed angles that intercept the same arc are congruent

  
• Segment of Secants and Tangents Theorem says the product of the outside segment and the whole secant equals the square of the tangent to the same point

  
• Corresponding Arcs Congruency Theorem says if two chords of a circle are congruent, then the corresponding arcs are also congruent

  
• The Perpendicular Bisector of a Chord Theorem is if a diameter of a circle is perpendicular to a chord, the diameter bisects the chord and arc

  
• The Segments of Chords Theorem says when two chords intersect within a circle, the product of the lengths of one chord's segments is equal to the product of the other chord's segments