Circle Geometry

Circle Geometry

1. Angle Theorems

   • Equal arcs on equal circles of equal radii subtend equal angles at the centre of the circle.
   • The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc.
   • The angle in a semicircle is a right angle.
   • Angles in the same segment are equal.

2. Tangent Theorems

   • Two circles touch if they have a common tangent at the point of contact.
   • The tangent to a circle is perpendicular to the radius drawn to the point of contact.
   • Tangents to a circle from an external point are equal.
   • When two circles touch, their centres and the point of contact are collinear.
   • The angle between a tangent and a chord drawn to the point of contact is equal to the angle in the alternate segment.

3. Chord Theorems

   • The perpendicular from the centre of a circle to a chord bisects the chord.
   • The line from the centre of a circle to the midpoint of a chord is perpendicular to the chord.
   • The perpendicular bisector of a chord passes through the centre of the circle.
   • Equal chords in a circle subtend equal angles at the centre.
   • Chords that are equidistant from the centre of the circle are equal.

4. Intersecting Secants & Chords

   • The products of the intercepts of two intersecting chords are equal.
   • The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point.
   • The products of the intercepts of two intersecting secants to a circle from an external point are equal.

5. Cyclic Quadrillaterals

   • The opposite angles of a cyclic quadrilateral are supplementary.
   • An exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
   • If the opposite angles in a quadrilateral are supplementary then the quadrilateral is cyclic.
   • If two points lie on the same side of an interval, and the angles subtended at these points by the interval are equal, then the two points and the endpoints of the interval are concyclic.