PQRS is a convex quadrilateral such that each of the four sides PQ, QR, RS and SP as well as each of the two diagonals PR and QS have rational lengths. The two diagonals intersect at the point N.
Will all the four line segments PN, QN, RN and SN have rational lengths?
Assume that PQRS is a trapezoid with PQ || RS. Then triangles PQN and RSN are similar with ratio PQ/RS. Then each diagonal is divided into two segments with the ratio PQ/RS. Since PQ, RS and both diagonals are all rational, then the segments PN, QN, RN and SN are also all rational.
Now, all that is left to prove are quadrilaterals without any parallel sides.