Circle: Encyclopedia II - Circle - Properties

Circle - Properties

Circle - Chord properties

• Chords equidistant from the centre of a circle are equal.
• Equal chords are equidistant from the centre.
• A line from the centre, perpendicular to a chord, bisects the chord.
• The line segment drawn from the centre to the midpoint of the chord is perpendicular to the chord.
• The perpendicular bisector of a chord passes through the centre of a circle.

Circle - Tangent properties

• The line drawn perpendicular to the end point of a radius is a tangent to the circle.
• A line drawn perpendicular to a tangent at the point of contact with a circle passes through the centre of the circle.
• Tangents drawn from a point outside the circle are equal in length.
• Two tangents can always be drawn from a point outside of the circle.

Circle - Inscribed angle theorem

• If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
• If two angles are inscribed on the same chord and on the same side of the chord , then they are equal.
• An inscribed angle subtended by a semicircle is a right angle.
• For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.

a line which touches the circle.

Circle - Secant tangent and chord properties

• The chord theorem states that if two chords, CD and EF, intersect at G, then . (Chord Theorem)
• If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then . (Tangent Secant Theorem)
• If two secants, DG and DE, also cut the circle at H and F respectively, then . (Corollary of the Tangent Secant Theorem)
• The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord. (Tangent Chord Property)
• If the angle subtended by the chord at the centre is 90 degrees then l = sqrt(2) * r, where l is the length of the chord and r is the radius of the circle.

Adapted from the Wikipedia article "Properties", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki