J.S.S public school,
JSS Public School
Introduction To Cyclic Quadrilaterals
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is
a quadrilateral whose vertices all lie on a single circle. This circle is called
thecircumcircle or circumscribed circle, and the vertices are said to be
concyclic. The center of the circle and its radius are called the circumcenter
and the circumradius respectively. Other names for these quadrilaterals are
concyclic quadrilateral and chordal quadrilateral, the latter since the sides
of the quadrilateral are chords of the circumcircle. Usually the quadrilateral
is assumed to be convex, but there are also crossed cyclic quadrilaterals.
The formulas and properties given below are valid in the convex case.
The word cyclic is from the Greek kuklos which means "circle" or "wheel".
All triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral
that cannot be cyclic is a non-square rhombus.
Properties of a Cyclic Quadrilateral
1. The opposite angles of a cyclic
quadrilateral are supplementary.
The sum of either pair of opposite
angles of a cyclic quadrilateral is 1800
2. If one side of a cyclic quadrilateral
are produced, then the exterior angle
will be equal to the opposite interior
3. If the sum of any pair of opposite
angles of a quadrilateral is 1800, then
the quadrilateral is cyclic.
Area of a Cyclic Quadrilateral
The area K of a cyclic quadrilateral with sides a, b, c, d is given by Brahmagupta's formula.
Where s, the semi perimeter, is
. It is a corollary to Bretschneider's formula since
opposite angles are supplementary. If also d = 0, the cyclic quadrilateral becomes a triangle
and the formula is reduced to Heron's formula.
The cyclic quadrilateral has maximal area among all quadrilaterals having the same sequence
of side lengths. This is another corollary to Bretschneider's formula. It can also be proved using
Four unequal lengths, each less than the sum of the other three, are the sides of each of three
non-congruent cyclic quadrilaterals, which by Brahmagupta's formula all have the same area.
Specifically, for sides a, b, c, and d, side a could be opposite any of side b, side c, or side d.
A cyclic quadrilateral with successive sides a, b, c, d and semi perimeter s has the
circumradius (the radius of the circumcircle) given by
This was derived by the Indian mathematician Vatasseri Parameshvara in the 15th century.
Using Brahmagupta's formula, Parameshvara's formula can be restated as
Theorems of Cyclic Quadrilateral
Cyclic Quadrilateral Theorem
The opposite angles of a cyclic quadrilateral are
An exterior angle of a cyclic quadrilateral is equal
to the interior opposite angle.
A D 1800
C B 180
Proving the Cyclic Quadrilateral Theorem- Part 1
The opposite angles of a cyclic quadrilateral
ABD ACD 360
B C 180 0.
ABD ACD 180
Thus,B + C = 180 .
Proving the Cyclic Quadrilateral Theorem- Part 2
An exterior angle of a cyclic quadrilateral is
equal to the interior opposite angle.
2 4 180
Opposite angles of a cyclic
4 5 180
Supplementary Angle Theorem
4 5 2 4
2 = 5.
Thus,5 = 2.
Using the Cyclic Quadrilateral Theorem
Finally we conclude that this given PPT on Cyclic
Quadrilaterals was very helpful, educational, and was fun
too. So we thank our mathematics teacher for giving us
this PPT assignment. While creating this PPT we had a
great time while doing it and while sharing our ideas.