Triangle categorizer -- completely refactored and re-written
A few simple comments:

Spelling:

Triangles are either Isosceles, Equilateral, or Scalene

Also instead of using string literals, replace them with macros like so:

#define EQUILATERAL "equilateral"
#define ISOSCELES "isosceles"
#define SCALENE "scalene"


I commend you on using long names in c, as many c-programmers use 1-2 letter names for things. I might suggest adding comments to complex code, such as mentioning the fact that you are using triangle inequality in your is_valid_triangle function

I would also suggest extending this program to work with non-integers, such as long longs, doubles and the like. In addition, it may be more helpful to a user to pass in coordinates instead of side lengths. That would also allow you to not check if a triangle is valid or not, as technically all 3 point pairs form a triangle (even if it is degenerate).