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that satisfy 
be a sequence defined by 
. Then the product of four consecutive terms in can be written as the product of two terms in . Find 
if 
.
with 
, 
and area 
. The circle 
with diameter 
meets 
at 
. Let the orthocenter of the triangle be 
, connect 
and extend to meet at 
such that 
and 
is the diameter of . Draw the circumcircle 
of 
, chord 
of is tangent to and it passes through , compute .
and 
Prove that
Let and 
Prove that
Let and 
Prove that
Let and 
Prove that
with 
. The points 
lie on 
respectively, such that 
. Extend 
to meet the circumcircle of at 
respectively. Let the circumcircles of 
meet at points 
. Extend 
to at 
, given 
, find ![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
.
with 
, denote the incenter of the triangle as 
. Extend 
to meet the circumcircle of 
at 
, find the length of 
.
and 
. Prove that
Equality holds when 
Equality holds when 
Equality holds when 
and 
. Then we should find this ratio: 
.![\[S_{EGC}=(a+x)^2+\frac{x^2}{2}-\frac{(a+x)(a+2x)}{2}-\frac{a(a+x)}{2}=\frac{x^2}{2}.\]](http://latex.artofproblemsolving.com/2/a/7/2a7271b3023b31cbdadac9161187b92f1a292986.png)
![\[S_{DEFBC}=(a+x)^2+x^2+\frac{ax}{2}.\]](http://latex.artofproblemsolving.com/c/f/0/cf022939edf4bba3c1ce126b096a1e193665deee.png)
![\[S_{DEFBC}=16S_{EGC}\Rightarrow a^2+2.5ax-6x^2=0\]](http://latex.artofproblemsolving.com/e/a/6/ea6e9085f8aac29b7c60539cc726c5be3c33a48e.png)
![\[\left(\frac{a}{x}\right)^2+2.5\frac{a}{x}-6=0,\quad \frac{a}{x}>0\Rightarrow\frac{a}{x}=\frac{3}{2}.\]](http://latex.artofproblemsolving.com/f/3/0/f3047f7e354ad1e790923424d1205422532724df.png)
Therefore,![\[\left(\dfrac{x}{a+x}\right)^2=\left(\dfrac{1}{\frac{a}{x}+1}\right)^2=\frac{4}{25}.\]](http://latex.artofproblemsolving.com/9/3/8/9388ead9972342deafa8461cdcb8961e91254ed0.png)
Hence answer is: 
.
lies on side 
and square 
is drawn outwards 
is 
of the area of pentagon 
.
