Is there any geometric demonstration that proves $m= \frac{y_2 -y_1}{x_2 - x_1}$ (the slope formula)? Is there any geometric demonstration that proves $$m= \frac{y_2 -y_1}{x_2 - x_1}?$$ 
I know one algebraic demonstration. It's:
1) If you have a function $y=mx+b$ and two points $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$, you can make two equations and solve for $m$: 
$$1) \: y_1=mx_1+b, $$
$$2) \: y_2=mx_2+b. $$
If you subtract equation $1$ from equation $2$, you get:
$$y_2-y_1=mx_2+b-(mx_1+b)$$
$$y_2-y_1=mx_2+b-mx_1-b$$
$$y_2-y_1=mx_2-mx_1$$
$$y_2-y_1=m(x_2-x_1)$$
$$ \frac{y_2 -y_1}{x_2 - x_1}=m.$$
That's the one I know, but since it's more of a geometry topic I'd rather like a geometric demonstration. Thanks.
I figured out this demonstration:



I am not an english speaker so I translated the best I could. Tell me what you think about it and if it´s valid. Thanks.
NOTE= In the second picture it should have said :"I´ll take a third point..."

Why did mathematicians decide that the slope was important?
It's important for a lot of reasons, but they mostly come down to differentiation. The wiki page has some beautiful geometrical demonstrations.
