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## 'Flexi Quads' printed from http://nrich.maths.org/

Use the scalar product of the diagonals. As the quadrilateral is
flexed the diagonals change but the lengths of the sides are
constant. All the vectors change but the the squares of the vectors
of the sides (representing the lengths) remain constant. To
preserve symmetry and obtain this scalar product in the form
required, write

$$2{\bf d}_1={\bf a}_1-{\bf a}_2-{\bf a}_3+{\bf a}_4, \quad
2{\bf d}_2=-{\bf a}_1-{\bf a}_2+{\bf a}_3+{\bf a}_4.$$