How is similarity of geometric figures applied and verified?

Two Geometric figures are similar if you can go from one to another only through rotation (spinning), translation(moving WITHOUT rotating), reflection (making a simetry) or scaling (like zooming)
When scaling, you have to scale both vertical and horizontal axis at the same rate (example, 2x, 7x or 0.3x)
As such, all squares are similar and so are all REGULAR poligons among each other (all equilateral triangles are similar, all regular pentagons, etc)
This gets a bit tricky if the polygons are not regular, but the principle remains the same. For example, if you have a triangle with sides measuring 2,5 and 4, it will be similar to a triangle with sides measuring 8, 20 and 16, because you have scaled it 4x, hence, it's internal angle remained the same.

Where are the following equations?