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by OronSH, Jul 31, 2024, 12:38 AM
Here's a proof that the Simson line of a point 
wrt 
bisects the segment connecting to the orthocenter.
First homothety at reduces it to showing that the Steiner line 
passes through the orthocenter. Now consider the parabola 
with focus and directrix . Then it is not hard to see that must be tangent to the sides of .
Let the perpendicular from
to 
intersect at 
. It is well-known that the tangents from to are perpendicular, since lies on its directrix.
Now use DDIT on point and the four lines 
and the line at infinity 
. Then is an inconic, and the tangents from are perpendicular. Next, 
and 
and by construction, the lines from to these points are perpendicular as well. Thus the involution must be the perpendicular involution.
This means the lines from to 
and 
are perpendicular as well, so 
and similarly 
and is the orthocenter as desired.
wrt 
bisects the segment connecting to the orthocenter.First homothety at
reduces it to showing that the Steiner line 
passes through the orthocenter. Now consider the parabola 
with focus and directrix . Then it is not hard to see that must be tangent to the sides of .Let the perpendicular from
to 
intersect at 
. It is well-known that the tangents from to are perpendicular, since lies on its directrix.Now use DDIT on point
and the four lines 
and the line at infinity 
. Then is an inconic, and the tangents from are perpendicular. Next, 
and 
and by construction, the lines from to these points are perpendicular as well. Thus the involution must be the perpendicular involution.This means the lines from
to 
and 
are perpendicular as well, so 
and similarly 
and is the orthocenter as desired.
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