Rectangular hyperbola objectively correct proof.
by qwerty123456asdfgzxcvb, Dec 12, 2024, 12:51 AM
Let a rectangular hyperbola intersect line 
at two points, 
. Two points are only conjugate in a circumconic of triangle 
if the circumconic passes through their cevapoint, by simple harmonic chasing. The polar of 
in a rectangular hyperbola is the line through its center through 
, as the pole of is its center by definition, and 
. Therefore 
are conjugate, so for any three points 
on the hyperbola, the circumconic must also pass through 
.
at two points, 
. Two points are only conjugate in a circumconic of triangle 
if the circumconic passes through their cevapoint, by simple harmonic chasing. The polar of 
in a rectangular hyperbola is the line through its center through 
, as the pole of is its center by definition, and 
. Therefore 
are conjugate, so for any three points 
on the hyperbola, the circumconic must also pass through 
.
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