Hi,

1) rotate the triangle AEX around the point E by 180 degrees and you'll obtain triangle DEX', as on the picture. Now DX' and CX are parallel (because DX' and AX are) and D is the midpoint of BC, so X' is the midpoint of BX (a basic result you should understand). Now since the length of XE is 1/2 of the length of XX' which is 1/2 of the lenght of XB, we have the length of XE is 1/4 of the length of XB.

As for 2) it would be polite to define what "two rectangles are equal" mean. If they mean that the rectangles have equal area, then since triangles ABP and QCP are similar, we have

BP/AB = CP/CQ

CQ.BP = AB.CP

AB.BP + CQ.BP = AB.BP + AB.CP

(AB+CQ).BP = AB.(BP+CP)

DQ.BP = AB.BC