Question

What's another name for an equilateral quadrilateral?

Answer 1

Equilateral quadrilateral is a rhombus.

Explanation:

Rhombus is usually defined as an equilateral parallelogram.
Since it's given that our quadrilateral is equilateral, all we have to prove is that it's a parallelogram.

Assume, we have a quadrilateral `ABCD` with all sides equal:
`AB=BC=CD=DA`

Draw a diagonal `AC` and consider two triangles `Delta ABC` and `Delta ADC`. They are congruent by "side-side-side" theorem.
Therefore the angles `/_ACD` and `/_CAB` are congruent.
Therefore, `AB` is parallel to `CD`.

Analogously, `AD` is parallel to `BC`. Hence, `ABCD` is a parallelogram. Together with length equality of its sides, it is sufficient for calling it a rhombus.

As a general theory of quadrilaterals refer to Unizor under menu items Geometry - Quadrangles (quadrangles and quadrilaterals are equivalent terms, one refers to four angles, another - to four sides).