I first observed that a/b is the tangent of angle DAE in Figure 1 and c/d the tangent of angle BDF.

I also see that the measure of angle BAC is greater than the measure of angle DAE.

All angles of interest for the problem are smaller than 90 degrees, and in this region the tangent is a monotonic increasing function (it does not decrease in this range), so the order relation between angles is transferred to tangents.

Now I write the tangents of these angles:

\tan DAE = \frac{DE}{AE} = \frac{a}{b}

\tan BAC = \frac{BC}{AC} = \frac{a + c}{b + d}

I have now to transpose the relation between the angles to their tangents, and I obtain the inequality

\frac{a}{b} < \frac{a + c}{b + d} for \frac{a}{b} < \frac{c}{d}