is defined recursively by ![\[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]](http://latex.artofproblemsolving.com/f/4/b/f4b8f95e49de7274cccd8ee498e0b496f31ddfea.png)
Show that 
for all 
.
be the circumcircle of a non-isosceles triangle 
be a tangent line to 
)
be midpoint of 
.
be its circumcenter and 
its incenter.
be the Gergonne point of the triangle 
is parallel with line 
.
such that, for all positive integers 
and 
,
Y by Mathuzb, buratinogigle, qwertyboyfromalotoftime, Adventure10, Mango247
Bisector of side 
intersects circumcircle of triangle 
in points 
and 
. Points 
and lie on the same side of line . Point 
is an orthogonal projection of point on line 
. Point 
is middle of line segment 
. Show that points 
lie on one circle.
intersects circumcircle of triangle 
in points 
and 
. Points 
and lie on the same side of line . Point 
is an orthogonal projection of point on line 
. Point 
is middle of line segment 
. Show that points 
lie on one circle.
as the unit circle and WLOG 
.
.
to be the unit circle and 
,
and 
.
,
bisects 
and 
,
of 
,
which is obvious using 
is the 
cuts 
is the reflection of 
are concyclic (lazy to use direct angle here)
so 
are collinear
by Reim theore
be the midpoints of 
is the simson line of 
; we have 
then 
hence to show that 
cyclic it suffices to show 
or 
but 
then the result follows .
be midpoint of 
= 
and 
= 
= 
,
= 
= 
= 
or 
= 
= 
,
= 
or 
= 
= 
= 
,
,
.
.
and 
.
is cyclic as desired.
and 
are similar.
and 
.
and 
are similar.
,
and 
.
and 
.
.
composed with the symmetry along the 
.
is an image of 
so that 
.
the image of 
.
.
.
,
,
