AwesomeMath 2016
by djmathman, Dec 27, 2015, 12:14 AM
I guess I'm going to help out with Cornell and Puget Sound (western site) this year~
and now for some geo (darn why am I still doing geo i'm in college nobody cares about geo anymore)
and now for some geo (darn why am I still doing geo i'm in college nobody cares about geo anymore)
Quote:
Points 
and 
are the circumcenter and orthocenter of acute triangle 
, respectively. The perpendicular bisector of segment 
meets sides 
and 
at 
and 
, respectively. Prove that 
and 
are the circumcenter and orthocenter of acute triangle 
, respectively. The perpendicular bisector of segment 
meets sides 
and 
at 
and 
, respectively. Prove that 
the circumcircle of 
.
be the antipode of 
with respect to 
and 
.
and 
.
and 
.
,
,
are collinear. Now let 
and 
.
This means 
is isosceles with 
.
,
is the midpoint of 
.
is the midpoint of 
,
.
with ratio of similitude 
as desired.![\[\angle AA_0E_0\equiv\angle AA_0B'=\angle ABB'=\angle ACC'=\angle AA_0C\equiv \angle AA_0D_0.\]](http://latex.artofproblemsolving.com/2/7/b/27ba0f877b92a333f155c1f078503b3100ffd0d6.png)
Finally, note that the homothety 
sends 
as desired. 
and 
are similar and so by spiral similarity, 
and 
are similar. This and it's vertex 
analog gives that 
and we are done.
April 2025
November 2024