Quadrilaterals inscribed in a circle and circumscribed about a circle

Designations:
R - radius of the circumscribed circle;
r - radius of the inscribed circle;
α, β, γ, δ - angles of the quadrilateral;
a, b, c, d - sides of the quadrilateral;
S - area of ​​the quadrilateral.

Quadrilaterals inscribed in a circle. Circles circumscribed around a quadrilateral.

A quadrilateral is called inscribed in a circle if the circle passes through all four vertices of the quadrilateral. The circle in this case is called circumscribed around the quadrilateral.

1. A quadrilateral can be inscribed in a circle if and only if the sums of the opposite angles are equal to 180 degrees: α+β = γ+δ = 180^∘.
2. Ptolemy's Theorem.
For an inscribed quadrilateral, the product of the lengths of the diagonals is equal to the sum of the products of the lengths of two pairs of its opposite sides AC*BD = AB*CD + BC*AD.
Converse theorem.
If the product of the lengths of the diagonals of a quadrilateral is equal to the sum of the products of the lengths of two pairs of its opposite sides, then such a quadrilateral can be inscribed in a circle.
3. A circle can be described
- about a parallelogram if and only if this parallelogram is a rectangle;
- about a rhombus if and only if this rhombus is a square;
- about a trapezoid if and only if the trapezoid is isosceles.
4. Brahmagupta's formula for the area of ​​an inscribed quadrilateral with semiperimeter p = (a+b+c+d)/2:

Quadrilaterals circumscribed around a circle. Circles inscribed in quadrilaterals.

A quadrilateral is called circumscribed around a circle if the circle touches all four sides of the quadrilateral. The circle in this case is called inscribed in the quadrilateral.

1. A quadrilateral can be described around a circle if and only if the sums of its opposite sides are equal a+c = b+d.
2. A circle can be inscribed
- into any rhombus;
- into any square.
3. If a circle can be inscribed in an isosceles trapezoid, then the side is equal to the midline.
4. The bisectors of all interior angles of the circumscribed quadrilateral intersect at one point - the center of the inscribed circle. 5. The formula for the area of a circumscribed quadrilateral through the radius of the inscribed circle and the sides of the quadrilateral:

S = (a+b+c+d)*r/2.

6. Formula for the area of a circumscribed quadrilateral if a circle can be described around it: