**Proof:**

Let's consider two triangles, triangle ABC and triangle DEF, such that two sides and a median bisecting one of the sides of triangle ABC are respectively proportional to the two sides and the corresponding medians of triangle DEF.

**Step 1: Establishing Proportions**

Let's denote the sides and medians of triangle ABC as AB, AC, and AD respectively. Similarly, the sides and medians of triangle DEF will be denoted as DE, DF, and DG.

Given that AB/DE = AC/DF = AD/DG.

**Step 2: Establishing Proportionality of Triangles**

To prove that the triangles are similar, we need to show that their corresponding angles are equal.

Let's start by comparing the sides AB and DE. From the given proportion, we have AB/DE = AC/DF. Cross multiplying this proportion, we get AB * DF = DE * AC.

Now, let's compare the sides AC and DF. From the given proportion, we have AC/DF = AD/DG. Cross multiplying this proportion, we get AC * DG = DF * AD.

**Step 3: Establishing Proportionality of Medians**

Next, let's compare the medians AD and DG. From the given proportion, we have AD/DG = AB/DE. Cross multiplying this proportion, we get AD * DE = DG * AB.

**Step 4: Establishing Proportional Angles**

Now, we can compare the ratios of the sides and medians to establish the proportionality of the angles.

Comparing AB * DF = DE * AC and AC * DG = DF * AD, we can conclude that the angles between the sides AB and AC are equal to the angles between the sides DE and DF.

Similarly, comparing AD * DE = DG * AB, we can conclude that the angles between the medians AD and AC are equal to the angles between the medians DE and DF.

**Step 5: Concluding Similarity**

Since the corresponding angles of the two triangles are equal, we can conclude that the triangles ABC and DEF are similar.

Therefore, if two sides and a median bisecting one of the sides of a triangle are respectively proportional to the two sides and the corresponding medians of another triangle, then the triangles are similar.