information:highlighting, structural:summarization
Diagonals

The lengths of the diagonals are

where a is the short base, b is the long base, and c and d are the trapezoid legs.

If the trapezoid is divided into four triangles by its diagonals AC and BD (as shown on the right), intersecting at O, then the area of △ {\displaystyle \triangle } AOD is equal to that of △ {\displaystyle \triangle } BOC, and the product of the areas of △ {\displaystyle \triangle } AOD and △ {\displaystyle \triangle } BOC is equal to that of △ {\displaystyle \triangle } AOB and △ {\displaystyle \triangle } COD. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.

Let the trapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and DC. Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD. Then FG is the harmonic mean of AB and DC:

The line that goes through both the intersection point of the extended nonparallel sides and the intersection point of the diagonals, bisects each base.