Isosceles, inscribed and circumscribed trapeziums

An isosceles trapezium has one pair of parallel sides (bases) and the other pair (legs) equal in length. For a trapezium to be inscribed in a circle (cyclic), the sum of the lengths of its two opposite sides must be equal: $AB + CD = BC + DA$. For a trapezium to be circumscribed about a circle (tangential), the sum of the lengths of its two pairs of opposite sides must be equal: $AB + CD = BC + DA$.

Important properties

• In an isosceles trapezium, the base angles are equal.

• A trapezium can be inscribed in a circle if and only if it is isosceles.

• A trapezium can be circumscribed about a circle if and only if the sum of the lengths of its two pairs of opposite sides are equal.

• In an isosceles trapezium, the diagonals are equal in length.