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be a cyclic quadrilateral. A circle centered at 
passes through 
and 
and meets lines 
and 
again at points 
and 
(distinct from 
). Let 
denote the orthocenter of triangle 
Prove that if lines 
are concurrent, then triangle 
and 
are similar.
such that 
, where 
are non-negative reals such that 
.
.
is determined by 
and the following recurrence relation for 
:![\[4u_{n+1} = \sqrt[3]{ 64u_n + 15.}\]](http://latex.artofproblemsolving.com/f/e/5/fe584783ca31a8ddf8cb2d7e1d18d110e3eaa406.png)
as 
given by ![$f(x) = \sqrt[3]{x + \dfrac {15} {64}}$](http://latex.artofproblemsolving.com/5/7/a/57a844d7180dd2ddcdca6ce73775ea8fbd6f3a97.png)
. It has one real root at 
, it is increasing, convex on ![$(-\infty,r]$](http://latex.artofproblemsolving.com/d/7/7/d7725672d062e8dbf42ca839cfbb73b349d4ad9d.png)
and concave on 
. The equation 
has three real roots at 
, 
and 
.
. The behaviour of as 
depends on the starting term , according to well-known techniques that resume to just considering the graph of 
and its intersection with the main bisector 
, and will be described by:
for 
; 
for 
; 
for 
.
