Question

Solve it please ,it's urgent ,

Answer 1

if the side AB~PQ , AC~PR and median AD~PM it means it all side are proportional to one
another then the triangle ABC~PQR

Answer 2

We can use Apollonius' theorem here.

Median AD through A on to BC: 
      AD = √[ (2AC²+ 2AB² - BC²)/4 ]

=>   BC² = 2 AC² + 2 AB² - 4 AD²    --- (1)

Given also: in ΔPQR

       QR² = 2 PQ² + 2 PR² - 4 PM²     --- (2)

We are given  AB/PQ = AC/PR = AD/PM = k (say)

 So  AB =k PQ ,  AC = k PR,  AD = k PM.

Substitute these in (1) to get:
 
      BC² = 2 k² PR² + 2 k² PQ² - 4 k² PM²
             = k² * QR²

=>   BC = k QR.

Since  AB/PQ = AC/PR = BC/QR, the two triangles are similar.
Proved.