Actually I should kick myself for not having finished this off earlier:

 

a²(p-q)(p-r)+b²(q-r)(q-p)+c²(r-p)(r-q) m² [(p-q)(p-r)+(q-r)(q-p)+(r-p)(r-q)] where m = min(a,b,c)

 

Now (p-q)(p-r)+(q-r)(q-p) = (p-q)(p-r+r-q) = (p-q)². Similarly

 

(q-r)(q-p) + (r-p)(r-q) = (q-r)² and

 

(r-p)(r-q)+(p-q)(p-r) = (p-r)²

 

Adding these three, we get  [(p-q)(p-r)+(q-r)(q-p)+(r-p)(r-q)] = 1/2 [(p-q)²+(q-r)²+(r-p)²]

 

Hence a²(p-q)(p-r)+b²(q-r)(q-p)+c²(r-p)(r-q)  m² 1/2 [(p-q)²+(q-r)²+(r-p)²]0

 

Only i have not used that they are sides of a triangle.