A cyclic quadrilateral is a quadrilateral that can be inscribed into a circle.

Variables

a,b,c,d
→
Sides
d1, d2
→
Diagonals
φ
→
Angle between diagonals
α, β, γ, δ
→
Interior angles
R
→
Radius of circumscribed circle
P
→
Perimeter
A
→
Area
L
→
Semiperimeter
1.
Sum of opposite angles
$\alpha + \gamma = 180\degree$
Cyclic quadrilateral
2.
Sum of opposite angles
$\beta + \gamma = 180\degree$
Cyclic quadrilateral
3.
Ptolemy's theorem
$ac+bd=d_1d_2$
Cyclic quadrilateral
4.
Permeter
$P = a+ b+ c+ d$
Cyclic quadrilateral
5.
Circumscribed radius
$R=\dfrac{1}{4} \sqrt{ \dfrac{(ac+bd)(ad+bc)(ab+cd)}{(p-a)(p-b)(p-c)(p-d)} }, \text{ where } p = \dfrac{a+b+c+d}{2}$
Cyclic quadrilateral
6.
Area
$A = \dfrac{1}{2} d_1d_2 \sin \psi$
Cyclic quadrilateral
7.
Area
$A = \sqrt{(p-a)(p-b)(p-c)(p-d)}, \text{ where } p=\dfrac{a+b+c+d}{2}$
Cyclic quadrilateral

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