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Trapezoid
 Bases of a trapezoid: $$a$$, $$b$$ Legs of a trapezoid: $$c$$, $$d$$ Midline of a trapezoid: $$m$$ Altitude of a trapezoid: $$h$$ Perimeter: $$P$$ Diagonals of a trapezoid: $$p$$, $$q$$ Angle between the diagonals: $$\varphi$$ Radius of the circumscribed circle: $$R$$ Radius of the inscribed circle: $$r$$ Area: $$S$$
1. A trapezoid (or a trapezium) is a quadrilateral in which (at least) one pair of opposite sides is parallel. Sometimes a trapezoid is defined as a quadrilateral having exactly one pair of parallel sides. The parallel sides are called the bases, and two other sides are called the legs.

2. A trapezoid in which the legs are equal is called an isosceles trapezoid. A trapezoid in which at least one angle is the right angle ($$90^\circ$$) is called a right trapezoid.

3. The midline of a trapezoid is parallel to the bases and equal to the arithmetic mean of the lengths of the bases.
$$m = {\large\frac{{a + b}}{2}\normalsize},\;\;m\parallel a,\;\;m\parallel b$$

4. Diagonals of a trapezoid (if $$a > b$$)
$$p = \sqrt {\large\frac{{{a^2}b - a{b^2} - b{c^2} + a{d^2}}}{{a - b}}\normalsize}$$,   $$q = \sqrt {\large\frac{{{a^2}b - a{b^2} - b{d^2} + a{c^2}}}{{a - b}}\normalsize}$$

5. Perimeter of a trapezoid
$$P = a + b + c + d$$

6. Area of a trapezoid
$$S = {\large\frac{{a + b}}{2}\normalsize} h = mh$$
$$S = {\large\frac{{a + b}}{2}\normalsize} \sqrt {{c^2} - {{\left[ {\large\frac{{{{\left( {a - b} \right)}^2} + {c^2} - {d^2}}}{{2\left( {a - b} \right)}}\normalsize} \right]}^2}}$$

7. All four vertices of an isosceles trapezoid lie on a circumscribed circle.

$$R = \large\frac{{c\sqrt {ab + {c^2}} }}{{\sqrt {\left( {2c - a + b} \right)\left( {2c + a - b} \right)} }}\normalsize$$

9. Diagonal of an isosceles trapezoid
$$p = \sqrt {ab + {c^2}}$$

10. Altitude of an isosceles trapezoid
$$h = \sqrt {{c^2} - \large\frac{1}{4}\normalsize{{\left( {a - b} \right)}^2}}$$

11. If the sum of the bases of a trapezoid is equal to the sum of its legs, all four sides of the trapezoid are tangents to an inscribed circle:
$$a + b = c + d$$

12. Radius of the inscribed circle
$$r = \large\frac{h}{2}\normalsize$$,
where $$h$$ is the altitude of the trapezoid.