It's not clear to me what sort of "expression" you want for the new height.

The original height h can be calculated in terms of a, b and c. I get the relation . If the sides of the base are changed from b,b,c to b,c,c, that is equivalent to exchanging b and c. So the new height k would be given by . You could then derive a (messy) expression for the ratio k/h if that is what you wanted.