Center of Rectangular Hyperbola on Nine-Point Circle
by AlastorMoody, Jun 1, 2019, 1:08 PM
Lemma: Let 
be rectangular hyperbola with a point 
on it and center 
. The reflection of over lies on
Problem: Let
be a triangle inscribed in a rectangular hyperbola 
. Prove, that the center of lies on the Nine-Point Circle WRT
Proof: Let be a fixed rectangular hyperbola through vertices of 
. If 
is the orthocenter WRT , then lies on . Let 
be the midpoints of 
. Let 
be the orthic triangle WRT . 
is the Nine-Point Circle WRT . Let the tangent to at 
; 
and 
intersect at 
, then 
intersect at which is the center of . Consider the reflections of 
over to 
. Then, lies on . Obviously 
is a parallelogram. Apply Pascal's Theorem on 
tangent to and the 
tangent to are parallel. Let 
, 
and 
. Then 
is the pedal triangle of 
WRT . Apply Pascal's Theorem on 
and 
is the Simson Line lies on 
lies on Nine-Point Circle. 
Remark: If is the fourth intersection of Circum-Rectangular Hyperbola of with , then Simson Line of passes through center of
be rectangular hyperbola with a point 
on it and center 
. The reflection of over lies on Problem: Let
be a triangle inscribed in a rectangular hyperbola 
. Prove, that the center of lies on the Nine-Point Circle WRT Proof: Let
be a fixed rectangular hyperbola through vertices of 
. If 
is the orthocenter WRT , then lies on . Let 
be the midpoints of 
. Let 
be the orthic triangle WRT . 
is the Nine-Point Circle WRT . Let the tangent to at 
; 
and 
intersect at 
, then 
intersect at which is the center of . Consider the reflections of 
over to 
. Then, lies on . Obviously 
is a parallelogram. Apply Pascal's Theorem on 
tangent to and the 
tangent to are parallel. Let 
, 
and 
. Then 
is the pedal triangle of 
WRT . Apply Pascal's Theorem on 
and 
is the Simson Line lies on 
lies on Nine-Point Circle. 
Remark: If
is the fourth intersection of Circum-Rectangular Hyperbola of with , then Simson Line of passes through center of This post has been edited 19 times. Last edited by AlastorMoody, Jun 4, 2019, 4:50 PM
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