Introduction:
In this problem, we are given two triangles ABC and PQR. We are told that the sides AB and AC of triangle ABC are proportional to the sides PQ and PR of triangle PQR, and the median AD of triangle ABC is proportional to the median BM of triangle PQR. We need to prove that triangle ABC is similar to triangle PQR.

Explanation:
To prove that two triangles are similar, we need to show that their corresponding angles are equal and their corresponding sides are proportional.

Step 1: Proving the angles are equal:
Since we are given that the sides AB and AC of triangle ABC are proportional to the sides PQ and PR of triangle PQR, we can conclude that angle BAC is equal to angle QPR. This is because the angles opposite to proportional sides in similar triangles are equal.

Step 2: Proving the sides are proportional:
We know that the median AD of triangle ABC is proportional to the median BM of triangle PQR. The medians of a triangle divide each other in a 2:1 ratio. Therefore, we can conclude that AD:BM = 2:1.

Now, let's consider the sides AB and AC of triangle ABC and the corresponding sides PQ and PR of triangle PQR. We are given that AB: PQ and AC: PR.

Since AD:BM = 2:1, we can rewrite AB as AD + DB and AC as AD + DC. Similarly, we can rewrite PQ as PM + MQ and PR as PM + MR.

Now, using the given proportions, we have:
(AD + DB): (PM + MQ) = AB: PQ (1)
(AD + DC): (PM + MR) = AC: PR (2)

From equation (1), we can rewrite AB: PQ as (AD + DB): (PM + MQ) = (AD: PM) + (DB: MQ).

Similarly, from equation (2), we can rewrite AC: PR as (AD + DC): (PM + MR) = (AD: PM) + (DC: MR).

Since AD:BM = 2:1, we can substitute AD: PM in the above equations with 2:1.

So, we have:
(2) + (DB: MQ) = AB: PQ
(2) + (DC: MR) = AC: PR

Since AB: PQ and AC: PR, we can conclude that DB: MQ = DC: MR.

Step 3: Conclusion:
From step 1, we know that angle BAC is equal to angle QPR.

From step 2, we know that the sides AB and AC of triangle ABC are proportional to the sides PQ and PR of triangle PQR.

Therefore, triangle ABC is similar to triangle PQR by the definition of similarity, which states that if two angles of one triangle are equal to two angles of another triangle, and the corresponding sides are proportional, then the triangles are similar.