newton gauss line objectively correct proof
by OronSH, Aug 30, 2024, 5:48 AM
In quadrilateral 
let 
Let 
be the midpoints of 
respectively. Then are collinear, and this line passes through the centers of all inconics of 
Proof: Let
denote the point at infinity along line 
DDIT at to complete quadrilateral 
gives that 
and 
are swapped. But are the lines through 
parallel to 
and since 
is the midpoint of 
line 
must be equally spaced in the middle of these two lines, so they are reflections over Similarly are reflections over
Thus the involution is reflection over so 
are reflections as well, thus the midpoint 
of 
lies on
Additionally the tangents from to any inconic are symmetric wrt its center, but they must also be reflections over and thus its center lies on this line.
let 
Let 
be the midpoints of 
respectively. Then are collinear, and this line passes through the centers of all inconics of 
Proof: Let
denote the point at infinity along line 
DDIT at to complete quadrilateral 
gives that 
and 
are swapped. But are the lines through 
parallel to 
and since 
is the midpoint of 
line 
must be equally spaced in the middle of these two lines, so they are reflections over Similarly are reflections over Thus the involution is reflection over
so 
are reflections as well, thus the midpoint 
of 
lies on Additionally the tangents from
to any inconic are symmetric wrt its center, but they must also be reflections over and thus its center lies on this line.
and 
into the orthocenter of 
– now 
May 2025
April 2025