As seen , what steps you have written you just want to prove if in a triangle two medians have common point of intersection , a point P, then the third median will also pass through point P.
The point of intersection of medians is called Centroid.
The statement ahead of :Statement: Point P lies on and .
Reason: from above algebra
Point P is the intersection of three medians which is called Centroid of triangle.The Centroid divides the line from opposite vertex in the ratio of 2:1.