A plane is a flat 2-dimensional surface that extends infinitely far in all directions. Typically a rectangular portion of a plane in represented in diagrams. Can you show an example where a plane is shown to be triangular in shape?

That is a totally unintelligible statement.
aamenis, if you cannot write a readable sentence, how do you expect to receive an understandable answer?
In any standard geometry course this is an axiom: Any three nonlinear points determine a plane.
Definitions & axioms do not have to make sense to you. You are not the arbitrator of what makes sense.
Your task is to understand the implications of the definitions & axioms.