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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)
.
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 
, denote the incenter of the triangle as 
. Extend 
to meet the circumcircle of 
at 
, find the length of 
.
such that 
.
as 
respectively, compute the area of 
given the area of 
.
![$[MDN] = \tfrac{1}{4}[DAC] = \tfrac{1}{4}(50-[ABC]) = 5$](http://latex.artofproblemsolving.com/6/5/e/65e7cd148a3a118fd5c4dac06bcebf2d87b3aeee.png)
.![$[ABM]+[BCN] = \tfrac{1}{2}([BAD]+[BCD]) = \tfrac{1}{2}[ABCD] = 25$](http://latex.artofproblemsolving.com/a/3/2/a3279ce95ccbf7e27943b67465a957553c99a518.png)
.![$[BMN] = [ABCD] - [MDN] - [ABM] - [BCN] = 50 - 5 - 25 = \boxed{20}$](http://latex.artofproblemsolving.com/2/2/8/228a8af298a76bd7cd046232a48733d74263254c.png)
.