Question 393628
No, there is no such formula. A counterexample would suffice.

Let h = 1, r = 3, and s = 2.
==> c = r + s = 5.
Then {{{a = sqrt(3^2 + 1^2) = sqrt(10)}}}, and
{{{b = sqrt(2^2 + 1^2) = sqrt(5)}}}.  Then 
{{{(as + br)/(r+s) = (as + br)/c = (2sqrt(10) + 3sqrt(5))/5 = 2.61}}}, to 2 decimal places.  But obviously this is not equal to h = 1.