xooks rbo
by OronSH, Feb 28, 2024, 2:19 PM
antigonal conjugates 
Let
be any point other than the orthocenter of 
.
Its antigonal conjugate
is the reflection of across the center of the rectangular hyperbola through 
, 
, 
, .
i. We have
and so on.
ii. The isogonal conjugates of and are inverses with respect to the circumcircle.
i. The Poncelet point of with respect to 
lies on the nine-point circle of 
however it is also the Poncelet point of with respect to 
Now, taking a homothety at with scale factor 
sends the nine-point circle of 
to the circle through 
where 
is the reflection of over 
Thus the antigonal conjugate of lies on this circle, so 
ii. Let
be the center of the rectangular circumhyperbola 
through 
and let 
be the two points at infinity on the hyperbola. Since 
we see that the cross ratio is equal to 
Under isogonal conjugation, are sent to the intersections of the isogonal conjugate 
of the rectangular circumhyperbola with the circumcircle. Now 
are sent to points on 
but since isogonal conjugation preserves cross ratio (for example, take the pencil through ), they must be sent to inverses.
Let
be any point other than the orthocenter of 
.Its antigonal conjugate
is the reflection of across the center of the rectangular hyperbola through 
, 
, 
, .i. We have
and so on.ii. The isogonal conjugates of
and are inverses with respect to the circumcircle.i. The Poncelet point of
with respect to 
lies on the nine-point circle of 
however it is also the Poncelet point of with respect to 
Now, taking a homothety at with scale factor 
sends the nine-point circle of 
to the circle through 
where 
is the reflection of over 
Thus the antigonal conjugate of lies on this circle, so 
ii. Let
be the center of the rectangular circumhyperbola 
through 
and let 
be the two points at infinity on the hyperbola. Since 
we see that the cross ratio is equal to 
Under isogonal conjugation, are sent to the intersections of the isogonal conjugate 
of the rectangular circumhyperbola with the circumcircle. Now 
are sent to points on 
but since isogonal conjugation preserves cross ratio (for example, take the pencil through ), they must be sent to inverses.This post has been edited 2 times. Last edited by OronSH, Feb 29, 2024, 9:38 PM
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