AoPS Academy
be an acute triangle with 
, and its vertices lie on the circle 
. Let 
be the altitude from vertex 
. Let 
and 
be the feet of the perpendiculars from 
to the lines 
and 
, respectively. Let 
intersect the circle again at points 
and 
such that lies between and . Let the lines and intersect at point 
. Let 
be the midpoint of segment . Let 
intersect line 
at point 
.
.
is parallel to line 
.
be the orthocenter of triangle 
, and let 
be the midpoint of segment . 
is the center (HBC). Let point 
lie on line 
such that ray 
is the angle bisector of 
. Prove that lines and 
intersect at a point lying on circle .
(i.e., the square on 
has as one of its sides and lies outside ). Show that the lines drawn from the vertices 
, 
, and 
to the centers of the opposite squares are concurrent.
is the amount of bananas Kyle currently has in his garage (where 
is in hours), then what does 
mean in this context with units or something.
hours, the rate at which the amount of Bananas Kyle has is decreasing by 
bananas per hour per hour)
. Prove that
Prove that
