Since I got tired of sucking at non-geo
by shiningsunnyday, Sep 20, 2016, 7:41 PM
1998 ISL G5 / Lemmas in Oly Geo wrote:
Let 
be a triangle with orthocenter 
circumcenter 
and circumradius 
Let 
be the reflections of the vertices 
across the opposite sides. Prove that they are collinear if and only if 
be a triangle with orthocenter 
circumcenter 
and circumradius 
Let 
be the reflections of the vertices 
across the opposite sides. Prove that they are collinear if and only if 
condition and a random collinearity?![[asy]
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[/asy]](http://latex.artofproblemsolving.com/a/a/b/aabdfff2b530baef75775cbcceda32c82d5b1cfa.png)
is defined as the midpoint of 
to work backwards to find our triangle whose Simson line is 
![[asy]
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[/asy]](http://latex.artofproblemsolving.com/8/a/2/8a21ec805e6916437f8daf58f3fdb08ec66f493a.png)
is the desired triangle whose Simson line with respect to point 
is 
Thus the collinearity condition is just another way of saying 
is cyclic. But why did I name the three vertices 
In fact, it is the case that 
since 
is defined as the line perpendicular to 
through 
etc.
it's not hard to see 
the centroid, is the center of the homothety.![[asy]
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[/asy]](http://latex.artofproblemsolving.com/7/7/1/771a1464af96af2a23165611fe8ec4b2c284acd7.png)
bisects 
and 
bisects 
it's easy to see that the B-median of 
also passes through the midpoint of 
Better, we see that the scale factor of the homothety is 
(using the fact the centroid divides a median into 
)
iff 
Well we do know that 
and so if only we can prove 
is in fact homothetic to 
then we we'll be done - 
is collinear.
with 
as the center of homothety, and so we're really just looking at the Euler line! Moreover, it's well known 
Since 
we have that 
and so 
implying This post has been edited 5 times. Last edited by shiningsunnyday, Oct 6, 2016, 11:41 AM
May 2017
April 2017