Rectangular hyperbola objectively correct proof.

by qwerty123456asdfgzxcvb, Dec 12, 2024, 12:51 AM

Let a rectangular hyperbola intersect line $IJ$ at two points, $\infty_1, \infty_2$. Two points are only conjugate in a circumconic of triangle $ABC$ if the circumconic passes through their cevapoint, by simple harmonic chasing. The polar of $I$ in a rectangular hyperbola is the line through its center through $J$, as the pole of $IJ$ is its center by definition, and $(I,J;\infty_1, \infty_2)=-1$. Therefore $I,J$ are conjugate, so for any three points $A,B,C$ on the hyperbola, the circumconic must also pass through $I \star J = H$.

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Additionally this should also get you that the crosspoint of $I,J$ is just $O$

by qwerty123456asdfgzxcvb, Dec 12, 2024, 12:53 AM

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coordbash
sorry abf

by EpicBird08, Dec 12, 2024, 2:21 AM

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@tarun coordbash proof is actually really simple lol bc it's just y=1/x

by qwerty123456asdfgzxcvb, Dec 12, 2024, 2:29 AM

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@above exactly

by EpicBird08, Dec 12, 2024, 4:22 AM

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JUST isogonal conjugate the hyperbola it has to have 90 degree angles to where it intersects the circumcircle . so . it apasses through the circumcenter.THEN isogonal conjugate bafkc. to get orthocenter

by OronSH, Dec 12, 2024, 2:07 PM

susus

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