# Formulas and Tables

Elementary Geometry# Quadrilateral

Internal angles: \(\alpha\), \(\beta,\) \(\gamma,\) \(\delta\)

Diagonals of a quadrilateral: \({d_1},\) \({d_2}\)

Angle between the diagonals: \(\varphi\)

Area: \(S\)

Radius of the inscribed circle: \(r\)

Perimeter: \(P\)

Semiperimeter: \(p\)

- A quadrilateral is a polygon with four sides and four vertices (angles).

- Types of quadrilaterals

A quadrilateral with two pairs of parallel sides is called a parallelogram.

A quadrilateral in which at least one pair of opposite sides is parallel is called a trapezoid.

A quadrilateral in which all four angles are right angles is called a rectangle.

A quadrilateral in which all four sides are equal is called a rhombus.

A quadrilateral in which all four sides are equal and all four angles are right angles is called a square. - Sum of the angles in a quadrilateral is \(360^\circ:\)

\(\alpha + \beta + \gamma + \delta \) \(= 360^\circ\) - Perimeter of a quadrilateral

\(P = a + b \) \(+\; c + d\) - Area of a quadrilateral

\(S = {\large\frac{1}{2}\normalsize}{d_1}{d_1}\sin \varphi ,\)

where \({d_1}\) and \({d_2}\) are the diagonals of the quadrilateral and \(\varphi\) is the angle between them. - If the sums of opposite angles in a convex quadrilateral are equal to \(180^\circ,\) then the quadrilateral is called a cyclic quadrilateral. All four vertices of a cyclic quadrilateral lie on a circumscribed circle.

\(\alpha + \beta = \gamma + \delta \) \(= 180^\circ\)

- Ptolemy’s theorem

In a cyclic quadrilateral, the sum of the products of opposite sides is equal to the product of the diagonals:

\(ac + bd = {d_1}{d_2}\) - Radius of the circumscribed circle

\(R =\) \({\large\frac{1}{4}\normalsize}\sqrt {\large\frac{{\left( {ac + bd} \right)\left( {ad + bc} \right)\left( {ab + cd} \right)}}{{\left( {p – a} \right)\left( {p – b} \right)\left( {p – c} \right)\left( {p – d} \right)}}\normalsize} ,\)

where \(p = {\large\frac{P}{2}\normalsize}\) is the semiperimeter of the quadrilateral. - If the sums of opposite sides in a convex quadrilateral are equal, then the quadrilateral is called a tangential quadrilateral. All four sides of a tangential quadrilateral are tangents to an inscribed circle.

\(a + d = b + c\)

- Radius of the inscribed circle

\(r =\) \({\large\frac{{\sqrt {d_1^2d_2^2 – {{\left( {a – b} \right)}^2}{{\left( {a + b – p} \right)}^2}} }}{{2p}}\normalsize},\)

where \(p = {\large\frac{P}{2}\normalsize}\) is the semiperimeter of the quadrilateral.