A Paralellogram Characteristic Problem

It is known a 4-gon ABCD. The midpoints of AB, CD, BD and AC are P, Q, R, S. Prove that PQ and RS intersects at mid points.

Answer.
The condition above can be drawn below!

Connect PQRS such that it is drawn a smaller 4-gon.

Look at triangle ABD and PBR!
Obvious BP:BA = BR:BD. So that we can conclude that triangle BPR similar to triangle BAD. Thus, we can conclude that PR // AD ……… (1*)

Look at triangle CAD and CSQ!
Obvious CS:CA = CQ:CD. So that we can conclude that triangle CSQ similar to triangle CAD. Thus, we can conclude that QS // AD ……… (2*)

Because PR // AD and QS // AD, then we conclude that PR // QS ……… (3*)

Look at triangle ABC and APS!
Obvious AP:AB = AS:AC. So that we can conclude that triangle APS similar to triangle ABC. Thus, we can conclude that PS // BC ……… (4*)

Look at triangle DRQ and DBC!
Obvious DR:DB = DQ:DC. So that we can conclude that triangle DRQ similar to triangle DBC. Thus, we can conclude that QR // BC ……… (5*)

Because PS // BC and QR // BC, then we conclude that PS // QR ……… (6*)

Because PR // QS and PS // QR, we can conclude that PQRS is a parallelogram by the definition. And its characteristics, by theorem 1.6, we can see that PQ and RS intersects at midpoint of it.