# Formulas

## Elementary Geometry

Sides of a quadrilateral: $$a,$$ $$b,$$ $$c,$$ $$d$$
Internal angles: $$\alpha$$, $$\beta,$$ $$\gamma,$$ $$\delta$$
Diagonals of a quadrilateral: $${d_1},$$ $${d_2}$$
Angle between the diagonals: $$\varphi$$
Area: $$S$$
Radius of the circumscribed circle: $$R$$
Radius of the inscribed circle: $$r$$
Perimeter: $$P$$
Semiperimeter: $$p$$
1. A quadrilateral is a polygon with four sides and four vertices (angles).
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.
3. Sum of the angles in a quadrilateral is $$360^\circ:$$
$$\alpha + \beta + \gamma + \delta$$ $$= 360^\circ$$
$$P = a + b$$ $$+\; c + d$$
$$S = {\large\frac{1}{2}\normalsize}{d_1}{d_1}\sin \varphi ,$$
where $${d_1}$$ and $${d_2}$$ are the diagonals of the quadrilateral and $$\varphi$$ is the angle between them.
6. 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.
$$\alpha + \beta = \gamma + \delta$$ $$= 180^\circ$$
7. Ptolemy’s theorem
In a cyclic quadrilateral, the sum of the products of opposite sides is equal to the product of the diagonals:
$$ac + bd = {d_1}{d_2}$$
8. 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.
9. 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.
$$a + d = b + c$$
10. 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.