Prove that “Equal chords of a circle
subtend equal angles at the centre”.
Prove that “Chords of a circle which subtends equal angles at the centre are equal”.
Prove that “The perpendicular from the centre of a circle
to a chord bisects the chord.”
Prove that “The line drawn through the centre of a circle to bisect a
chord is perpendicular to the chord”.
Prove that “Chords equidistant from the centre of a circle
are equal in length”
Prove that “Chords of a circle which are equidistant from the centre are equal”
Prove that “Of any two chords
of a circle then the one which is larger is nearer to the centre.”
Prove that “Of any two chords
of a circle then the one which
is nearer to the centre is larger.”
Prove that “line joining the midpoints of two equal chords of circle
subtends equal angles with the
chord.”
Prove that “if two chords of a circle bisect each other they must be diameters.
If two chords of a circle are equally inclined to
the diameter through their point of intersection, prove that the chords are
equal.
Prove that “The angle subtended by an arc at the centre is double the angle subtended by it at
any point on the remaining
part of the circle.”
Prove that “Angles in the same segment of a circle are equal.”
Prove that “Angle in a semicircle is a right angle.”
Prove that “Arc of a circle
subtending a right angle at any point of the circle in its alternate
segment is a semicircle.”
Prove that “Any angle
subtended by a minor arc in the
alternate segment is acute and any angle subtended by a major
arc in the alternate segment is obtuse.”
Prove that “If a
line segment joining two points
subtends equal angles at two
other points lying on the
same side of the line
segment, the four points are concyclic.”
Prove that “Circle drawn on any one
side of the equal sides of an isosceles
trainlge as diameter bisects the side.”
Prove that “The sum of either
pair of opposite angles of
a
cyclic quadrilateral is 180º.”
Prove that “If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is
cyclic.”
Prove that “If one
side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle.”
Prove that “If two
sides of a cyclic quadrilateral are parallel,
then the remaining two sides
are equal and the diagonals
are also equal.”
Prove that “If two
opposite sides of cyclic quadrilateral are equal, then the other two sides are parallel.”
Prove that “If two non parallel sides of a trapezium are equal, it is cyclic.”
Prove that “The sum
of the angles in the
four
segments exterior to
a cyclic quadrilateral
is
equal to 6 right angles.”
Two circles with centres
A and B intersect at C and D. Prove that ÐACB = ÐADB.
Bisector AD
of AC of DABC passes through the centre of the circumcircle of DABC. Prove
that AB = AC.
The diagonals of a cyclic
quadrilateral are at right angles.
Prove that perpendiculars from the point of their intersection on any side when produced backward bisect the
opposite side.
If
two circles intersect at two points, prove that their centres lie on the
perpendicular bisector of the common chord.
If
two intersecting chords of a circle make equal angles with the diameter passing
through their point of intersection, prove that the chords are equal.
Two
circles of radii 5 cm and
3 cm intersect at two points and the
distance between their centres is 4 cm. Find the length of the common chord.
If
two equal chords of a circle intersect within the circle, prove that the
segments of one chord are equal to corresponding segments of the other chord.
If
two equal chords of a circle intersect within the circle, prove that the line
joining the point of intersection to the centre makes equal angles with the
chords.
