AoPS Academy
be an acute angled triangle with orthocenter 
and 
. Let 
be any other point on the arc 
of the circumcircle of 
and let line 
intersect line 
at 
and let line 
intersect line 
at 
. Let the circumcircles of 
and intersect again at 
(
). Let the lines 
intersect the circumcircle of 
again at 
(
). Prove that the lines 
concur.
have altitudes 
circumcenter 
and orthocenter . Let 
be a point on line 
. The points 
are on lines 
respectively such that 
and 
. Prove that 
is perpendicular to 
.
is a triangle with incentre 
. The feet of the altitudes from to 
are 
respectively, and the line through 
parallel to 
intersects 
and 
at 
and 
respectively. Prove that the circles with diameters 
and 
have a common point on the circumcircle of 
is a triangle with incentre . The feet of the altitudes from to are respectively, and the line through parallel to intersects and at and respectively. Prove that the circles with diameters and have a common point on the circumcircle of 
Define 
and 
See that 
and 
swap under incircle inversion. Also, swaps with the nine-point circle of 
. Hence, 
(sharky-devil)
be the 
-Antipode in 
. Applying Pascal's Theorem on 
we get 
, since 
. Thus, we have 
. Similarly, 
so we are done.
