AoPS Academy
+3 w
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to 
, and let 
be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
and 
in the plane and a subset 
of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair 
of cities, they are connected with a road along the line segment if and only if the following condition holds: distinct from 
and 
, there exists 
such[/center]
is directly similar to either 
or 
.[/center]
is directly similar to 
if there exists a sequence of rotations, translations, and dilations sending 
to , 
to , and 
to 
.
is also the angle bisector of 
, which follows because 
. Hence, if we reflect across , our new point 
will lie on line 
and therefore satisfy 
. This is enough to imply that 
is equilateral, so 
. Therefore, 
.
, we can use strong Ptolemy's to finish. Alternatively, muscle memory gives that 
, and by Ptolemy we obtain 
, which is our final answer.
(i dont remember exactly if it was 8032038 or some other number, so correct me on this)
(i posted this at 12:29) lol
?
and 
with 
.
