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( 1)∫A( 1，23)、D(3，23)，

∴AD∥OC，

OA= 13, DC= 13 can be obtained from the distance formula between two points.

∴OA=DC.

AD = 2，OC=4，

∴AD≠OC

∴ trapezoid AOCD is isosceles trapezoid;

(2)∵M is the midpoint of AD,

∴AM=DM= 1，

∴M(2,23)，

Mo = MC = 4 can be obtained from the distance formula between two points.

OC = 4，

∴OM=OC=MC=4

∴△OMC is an equilateral triangle,

∴∠MOP=∠QCP=60。

∫∠MPQ = 60，

∴∠ 1+∠2=∠ 1+∠3= 120

∴∠2=∠3,

∴△OMP∽△CPQ

∴ OMPC = OPCQ, which means 4x = 4? x4？ y

∴y= 14x2？ x+4(0≤x≤4)；

(3)∵y= 14x2？ x+4，

∴y= 14(x？ 2)2+3,

When x = 2, the minimum value of y =3, that is MQ = 3.

When OP=3, x= 1, y= 134, that is, MQ= 134,

∴ When 0≤x≤2, the moving path length of point Q is 4-3= 1.

When 2 < x ≤ 3, the moving path length of point Q is 134? 3= 14

When point P moves from point O to point E (3 3,0), the path length of point Q is 1+ 14 = 54 units.