All quadrilaterals have intersecting diagonals.
Some symmetric quadrilaterals have perpendicularly bisecting diagonals.
Equilateral parallelograms
of which the square is a special case...
and the kite where two pairs of adjacent sides are the same length has one diagonal perpendicularly bisected by the other
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rhombus, kite, square
Squares, Rectangles, and Rhombuses.
trapezium
square
they are called parallelogram in which diagonals are equal