What do you call this method of proof?

[314Jason]

I'm trying to prove a statement using as many different methods as posible. I heard long ago that this is a type of proof, but I don't know what to call it!

For example: In a triangle ABC denoted by (equation with variable n), as n increases, the length of BC decreases.

So I've used mathematical/full induction to show how the distance decreases. Now I want to also show that as n increase the angle BAC decreases. And if angle BAC decreases, using logic, the length of BC must decrease as well.

And same goes for the area. I've proved that when n increases the area of triangle ABC decreases, which corresponds to the decreasing length of BC.

What do I call this type of proof? Is it analytical proof? :(

 

[nucl34rgg]

Well normally when you are using n as an argument and showing it is true for all n in the natural numbers greater than k starting from some particular natural number k, you are doing a proof by induction. However, in your case it seems to me that you are relating n with an angle in some way. Since the angle is defined over the reals mod 2pi, you would need to be careful about trying to use induction to prove this. I would say try to use geometric and analysis arguments to prove your statements. You know how to relate angles to lengths using trigonometry, and so you have bounds on these lengths. You could show how the lengths change as a function of the angle and try to show what happens when the angle approaches some value.