Cute Geometry Proof

Prove:  Any line L will cut a circle at most 2 points:

Let circle C (x,y) be unit circle defined by

C(x,y) : x² + y² = 1

 

Factorize C(x,y) : (x+iy) (x-iy) = 1 in the complex plane.
So C  = {L1} U {L2}
where L1 and L2 are two lines

L1= x+iy

L2= x - iy

L1 and L2 intersect at origin (0,0):

x+ iy = x-iy

We know that any line L will cut L1 at most 1 point, and L2 at most 1 point

Therefore,

L cuts the circle C at most (1+1=) 2 points. [QED]