AoPS Academy
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 .
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 
.
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)
.
lie inside a square 
such that the two triangles 
and 
are equilateral. Show that 
is an equilateral triangle.
![[asy]
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);
label("$A$",(0,4),NW); label("$B$",(4,4),NE); label("$C$",(4,0),SE); label("$D$",(0,0),SW);
label("$F$",(2,4-2*sqrt(3)),S);
draw((0,4)--(2,4-2*sqrt(3))--(4,4));
label("$E$",(4-2*sqrt(3),2),W);
draw((4,4)--(4-2*sqrt(3),2)--(4,0)); [/asy]](http://latex.artofproblemsolving.com/4/d/2/4d24a317f89c2381ce9435de01132258a0344156.png)
and 
.
and 
are equilateral, we find 
and 
.
.
.
,
is equilateral, as desired.
is equilateral, and 
.
.
,
.
.
,
,
.
.
,