be an acute triangle inscribed in a circle 
with orthocenter 
.
perpendicular to 
intersects 
at 
.
intersects 
at 
.
intersect 
across 
.
intersects 
different from 
.
and 
intersect on the circumcircle of triangle 
.
and 
is a circle tangent to 
The tangent
to 
If 
are the contact points of 
respectively, prove that 
such that 
for all 
be a triangle such that 
.
be a point on 
such that 
and let 
be the line passing through 
.
and let 
be the tangent line of 
.
is tangent to 
.
be the projection of 
.
bisects 
.
Y by
Let 
be an acute-angled triangle with 
. The altitudes 
and 
intersect at 
. Let 
be the midpoint of 
. Point 
is chosen on the extension of 
beyond and point 
is chosen on the segment 
such that 
. Prove that points 
and are collinear.
be an acute-angled triangle with 
. The altitudes 
and 
intersect at 
. Let 
be the midpoint of 
. Point 
is chosen on the extension of 
beyond and point 
is chosen on the segment 
such that 
. Prove that points 
and are collinear.
and we are done.
