Is there any tricks to memorize the special right triangle's sides?

No, but there is a way of creating them. If you let 
a = 2xy
b = x^2 - y^2 where x > y
c = x^2 + y^2 

And a^2 + b^2 = c^2 You can create these sides to your needs. For example

x = 5
y = 2

2xy = 2*5*2 = 20
x^2 - y^2 = 25 - 4 = 21
x^2 + y^2 = 25 + 4 = 29  which is not well known but it is a right angle.

29^2 = 20^2 + 21^2

841 = 400 + 441
841 = 841

Just one further comment. It is easy to check to see if the triangle you are talking about is a right triangle. Just use a^2 + b^2 = c^2. 

Generally it is unnecessary to either create your own or memorize ones you have come across. 

I think any math teacher would say "why bother?"
If you are not satisfied, put a reason for doing this in the comments. 

I’m confused what’s happening

Add up all the sides