,
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
.
chessboard into 
non-overlapping rectangles subject to the following conditions:
is the number of white squares in the 
-
.
.
denote the determinant of a square matrix 
.![$g:\left[0, \frac{\pi}{2}\right] \rightarrow R$](http://latex.artofproblemsolving.com/5/3/4/534fcbbc20e62de2dfa2579cd0642663c9fc3380.png)
be the function defined by
where 
.
be a quadratic polynomial whose roots are the maximum and minimum values of the function 
and 
.
Y by Adventure10, Mango247
Let 
be a triangle with 
. Let 
and 
be on the lines 
and 
, respectively, such that 
, 
, and 
, and lie on the same side of 
. Let 
be the incenter of triangle , and let 
be the orthocenter of triangle 
. Show that , , and are collinear.
be a triangle with 
. Let 
and 
be on the lines 
and 
, respectively, such that 
, 
, and 
, and lie on the same side of 
. Let 
be the incenter of triangle , and let 
be the orthocenter of triangle 
. Show that , , and are collinear.This post has been edited 3 times. Last edited by math163, Nov 10, 2018, 8:32 AM
let 
be the anticomplementary triangle and 
is the Nagel point of 
are the orthocenters of 
,
are concurrent at 
and now note that 
,
meet 
at 
,
,
so 
are isotomic conjugates wrt 
so we conclude 
are collinear 
