Doing Euler with my ruler!
by shiningsunnyday, Jul 5, 2016, 1:18 PM
2004 APMO P2 wrote:
Let
be the circumcenter and
the orthocenter of an acute triangle
. Prove that the area of one of the triangles
,
and
is equal to the sum of the areas of the other two.






Keeping the base-height area formula in mind, note how the three triangles share
as the base, and thus it suffices to prove
where
denotes the distance between
and Euler's line.
is impossible to compute each on their own, but note how we can drop an altitude from the midpoint
of
onto Euler's line and obtain a segment with length 
![[asy]
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[/asy]](//latex.artofproblemsolving.com/6/9/c/69c8af108110cf2fcc32818c60450ca2fa69fb54.png)
Fantastic, so now it suffices to show
where we let
be the foot of the latitude from
to Euler's line. The
suspiciously reminds us of
which lies on the intersection of
and
Now we're home.
is the image of the homothety of
with scale factor
and center
and thus
and the result follows.








![[asy]
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Fantastic, so now it suffices to show












Tidbit
Darn I'm such a noob I can't even prove the trigonometric equivalent of this problem. RIP. Well I'll be finishing all the problems in section 1 of EGMO soon. Time to learn to bash.
Tidbit 2
Also, AP score came out today. I got a 2 on my APUSH exam, and all my classmates got 5's. What a great way to end the day. Good night!
This post has been edited 6 times. Last edited by shiningsunnyday, Jul 5, 2016, 2:40 PM