In a triangle ABC, the points D, E and F are midpoints of the sides.
Compare the two triangles ABC and AFE. The sides AF || AB and AE || AC. The angle A is common. The ratio of sides AF/AB = AE/AC = 1/2. Then the two triangles AEF and ABC are similar (due to side-angle-side ratio property).
The angles AFE and ABC are equal. Then AEF and ACB are also equal. Hence, EF and AB are parallel and EF = AB/2.
Let the medians BE and CF intersect at O. We need to prove that AO and OD are parallel.
Look at the triangles OGE and ODB.
OB || OE and GE || BD . The included angles GOD and DOB (vertically opposite angles) are same. The angles GEO and EBD are equal, alternate angles between two parallel lines. Similarly, the angles OGE and ODB are equal (alternate angles).
The two triangles are similar.
Hence, OD || OG. ie., OD || OA. Hence proved.