Prove that the sum of the two medians of the triangles is greater than the half-sum of the two

Prove that the sum of the two medians of the triangles is greater than the half-sum of the two sides to which they are drawn.

To prove this statement, we need the medians in the triangle.

Let’s mentally imagine a triangle ABC and draw medians in it mentally.

Let’s call them A1A and B1B. Mentally, the point of their intersection will be O.

Then we have the following situation:

AO + OB1> AB1.

And:

BO + OA1> A1B.

From this data, we can infer the following:

AO + OB1 + BO + OA1> AB1 + BA1.

Well, in turn, from here we can safely deduce the following:

AA1 + BB1> 0.5 (AC + BC).

Thus, we have proved that the sum of the 2 medians is greater than the half-sum of the 2 sides.