ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the mid points of AB, AC, CD and BD respectively, show that PQRS is a rhombus.

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Given: ABCD is a quadrilateral in which AD = BC. P, Q, R and S be the midpoints of AB, AC, CD and BD respectively. 
To prove: PQRS is a rhombus.
Proof: By a theorem, line segment joining the mid points of any two sides of a triangle is parallel to the third side and equal to half of it.
In Δ ABC, PQ||BC and PQ = BC = DA
In Δ CDA, RQ||DA and RQ = DA
In Δ BDA, SP||DA and SP = DA
In Δ CDB, SR||BC and SR = BC = DA
Therefore SP || RQ, PQ || SR and PQ = RQ = SP = SR.
Hence PQRS is a rhombus.