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]
Mathtuition88 