Parallelogram

A parallelogram is a quadrilateral whose opposite sides are pairwise parallel.

Properties of a parallelogram

1. In a parallelogram, opposite angles and opposite sides are pairwise equal:
2. The sum of the one-sided angles of a parallelogram is 180°.
3. The diagonals of a parallelogram intersect and are divided in half by the point of intersection:

Signs of a parallelogram

1. If a quadrilateral has two parallel and equal sides, then the quadrilateral is a parallelogram.
2. If in a quadrilateral the opposite sides are pairwise equal, then this quadrilateral is a parallelogram.
3. If in a quadrilateral the opposite angles are pairwise equal, then this quadrilateral is a parallelogram.
4.If the diagonals of a quadrilateral are divided in half by their intersection point, then the quadrilateral is a parallelogram.

Varignon's theorem

A quadrilateral whose vertices coincide with the midpoints of the sides of an arbitrary quadrilateral is a parallelogram whose sides are parallel to the diagonals of the original quadrilateral.
The perimeter of this parallelogram is equal to the sum of the diagonals of the original quadrilateral, and the area of the parallelogram is equal to half the area of the original:

P_▱EFGH = AC + BD,    S_▱EFGH = S_▱ABCD / 2.