a^2 + b^2 / c^2 <=M (=1)

 a^2 + b^2    (1)
 ---------
     c^2

Law of cosines, replace C in terms of A and B and Theta...

c^2 = a^2 + b^2 - 2ab cos(theta) (2)

theta = 0; cos -> 1 (so subst with 1 early, reorder) theta = 180; cos-> -1

c^2 = a^2 - 2ab + b^2    (3)
c^2 = a^2 + 2ab + b^2`   (3n)
 
c^2 = (a-b)^2   (4)
c^2 = (a+b)^2   (4n)

add dual number.

c^2  = ( (a-b)+e)^2 )    (5)
c^2  = ( (a+b)-e)^2 )    (5n)

c^2 = (a-b)^2 +2(a-b)e + 0   (6)
c^2 = (a+b)^2 -2(a+b)e + 0   (6n)

reduce top part....

a^2 + b^2 = (a+b)(a-b)    (7)

Reassemble

(a+b)(a-b) / (a-b)^2 + 2(a-b)e   (8)  ( 7/6 )
(a+b)(a-b) / (a+b)^2 - 2(a+b)e   (8n)  ( 7/6n )
(a+b)(a-b) / (a-b)^2 + 2(a+b)e   (8n - normalize, use 8)

remove (a+b) from 8n

 if the other path is taken, it's infinite... (a+b)/(a-b) approaches infinity at A=B > 0

(a-b) / (a+b) + 2e   (9n)

this is a plot of (a-b) / (a+b) + (2*0) and (b-a) / (a+b) + (2*0)

https://www.desmos.com/calculator/by1chyufxa

And near 0 x, * small b, M approaches inifnity... or is < 0. at large A and B it's 1. at small C it's infinity (unless you look at it as c = a+b - e or c = a-b + e (or b-a)

d3x0r/Math.md