Let the sides of the large triangle be a, b, and c.
Let the distances to the vertices from an interior point 
be x, y, and z.

Half the perimeter of the large triangle is 




Use the triangular inequality on each of the three smaller triangles 
that make up the large triangle:

x + y     > a
x     + z > b
    y + z > c

Adding those three inequalities term by term:

 x +  y      > a
 x      +  z > b
      y +  z > c
--------------------
2x + 2y + 2z > a+b+c

Dividing through by 2

 x +  y +  z >  

Edwin