I'm looking for a more declarative way of stating an equation holds the below property:
$$\begin{align} 5a + 5b + 5c & \equiv a + b + c \\ 4a + 2b + 2c & \equiv 2a + b + c \\ 15a + 3b + 9c & \equiv 5a + b + 3c \\ 18a + 27b + 36c & \equiv 2a + 3b + 4c \\ & \dots \\ xa + yb + zc & \equiv \frac{x}{n}a + \frac{y}{n}b + \frac{z}{n}c \end{align}$$
Where $n$ is the largest common factor between $a, b, c$.
In other words, the ratio of each coefficient is important whereas the actual coefficients can be arbitrary. For what it's worth, $a, b, c$ may or may not equal each other. The equation deals with equivalence, not equality.
$$\begin{align} 5:5:5 & \equiv 1:1:1 \text{, } n = 5 \\ 4:2:2 & \equiv 2:1:1 \text{, } n = 2 \\ 15:3:9 & \equiv 5:1:3 \text{, } n = 3 \\ 18:27:36 & \equiv 2:3:4 \text{, } n = 9 \end{align}$$
What is the mathematical concept behind this, or how could I explain this concept more declaratively/formally?
Projective? Specifically, tuples of numbers with same proportion define a single projective point. These tuples are called homogeneous coordinate or projective coordinate of that projective point.