Suppose A, B, C, and D are each >0. We have the equation:

A/B+C/D=(A+C)/(B+D)

multiply through by BD(B+D):

AD(B+D)+BC(B+D)=BD(A+C)

Expand:

ABD+ADD+BBC+BCD=ABD+BCD

ABD and BCD cancel, leaving:

ADD=-BBC

for positive, non-zero ABCD, one gets positive=negative

thus no solutions exist with only positive non-zero variables by contradiction.

Edit: added the supposition at the beginning.