Properties of Circles

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1.���� Perpendicular Bisector Theorem

������� i.����� Every point on the perpendicular bisector of a line segment is equidistant from the end-points of the segment.

������� ii.���� If a point is equidistant from the end-points of a line segment, it lies on the perpendicular bisector of the segment.

��������������� ( ^ bisector theorem )

 

2.���� Chord Theorem

������� i.����� The centre of a circle lies on the perpendicular bisector of a chord of the circle.

��������������� ( ^ bisector of chord passes through centre )

������� ii.���� The perpendicular from the centre of a circle to a chord bisects the chord.

��������������� ( ^ from centre bisects chord )

������� iii.��� The line joining the centre of a circle and the mid-point of a chord, which is not a diameter, is perpendicular to the                chord.

��������������� ( line joining centre and mid-point of chord ^ chord )

������� iv.��� In the same circle or equal circles, chords equidistant from the centre are equal in length.

��������������� ( chords equidistant from centre are eq. )

������� v.���� In the same circle or equal circles, any two equal chords are equidistant from the centre.

��������������� ( eq. chords, equidistant from centre )

 

3.���� Angle Theorem

������� i.����� An angle of circumference subtended by an arc is half the central angle of the same arc.

��������������� ( � at centre = 2 � at circumference )

������� ii.���� An angle of circumference subtended by a semi-circle is a right angle.

��������������� ( � in semi-circle )

������� iii.��� Angles of circumference subtended by the same arc are equal.

��������������� ( �s in same segment )

������� iv.��� Opposite angles of a cyclic quadrilateral are supplementary.

��������������� ( opp. �s of cyclic quad. )

������� v.���� An exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

��������������� ( ext. � of cyclic quad. )

 

4.���� Tangent Theorem

������� i.����� A tangent to a circle is perpendicular to the radius at the point of contact.

��������������� ( tangent ^ radius )

������� ii.���� A line perpendicular to a radius of a circle at its outer endpoint is a tangent to the circle at this end-point.

��������������� ( converse of� tangent ^ radius )

������� iii.��� If two tangents are drawn to a circle from an external point, then

��������������� a)���� the tangent segments from this point to the points of contact are equal;

��������������� b)���� the tangent segments subtend equal angles at the centre;

��������������� c)���� the tangent segments make equal angles with the straight line joining the external point to the centre.

��������������� ( tangents from ext. pt. )

 

5.���� Tangent-chord Theorem

������� A tangent-chord angle of a circle is equal to an angle in the alternate segment of the circle.

��������������� ( � in alt. segment )

 

6.���� Concyclic Points Theorem

������� i.����� If the line segment joining two points subtend equal angles at two other points on the same side of it, then the four                 points are concyclic.

��������������� ( converse of �s in same segment )

������� ii.���� If the hypotenuse of a right triangle is a diameter of a circle, then the vertex of the right angle is a point of the circle.

��������������� ( converse of semi-circle )

������� iii.��� If a pair of opposite angles of a quadrilateral are supplementary, then its vertices are concyclic.

��������������� ( opp. �s of quad. supp. )

������� iv.��� If an exterior angle of a quadrilateral is equal to its interior opposite angle, then the quadrilateral ����

��������������� is cyclic.

��������������� ( ext. � of quad . eq. )

 

7.���� Arc-angle-chord Theorem

������� In the same or equal circles,

������� i.����� if two arcs subtend equal angles at the centre or at the circumference, the arcs are equal.

��������������� ( eq. � at centre or at circumference, eq. arcs )

������� ii.���� if two arcs are equal, they subtend equal angles at the centre or at the circumference.

��������������� ( eq. arcs, eq. � at centre or at circumference )

������� iii.��� arc length are proportional to the angles which they subtend at the centre or at the circumference.

��������������� ( arc length �� at centre or at circumference )

������� iv.��� if two chords are equal, the arcs cut are equal.

��������������� ( eq. chords, eq arcs )

������� v.���� if two arcs are equal, the corresponding chords are equal.�

��������������� ( eq arcs, eq chords )