,
,
and 
be four points on a line, in that order. Let 
be one of the intersections of the circle with diameter 
and 
are three points, I will call their circumcircle 
.
and 
are tangent, that 
and 
are tangent, and that the first two circumcircles are orthogonal to the last two circumcircles.
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 
.
its incenter.
be the isogonal conjugate of 
be the Gergonne point of the triangle 
is parallel with line 
.
Y by
Let 
be an acute-angled triangle with altitudes 
and 
. Points 
and 
are symmetric points of 
wrt lines 
and 
respectively. The circle circumscribed around 
has center 
and intersects the line for second time at point 
. Prove that 
.
be an acute-angled triangle with altitudes 
and 
. Points 
and 
are symmetric points of 
wrt lines 
and 
respectively. The circle circumscribed around 
has center 
and intersects the line for second time at point 
. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Jul 14, 2021, 10:23 AM
meet 
.
are homothetic and result is no surprize.
