be the circumcircle of right-angled triangle 
)
intersects the line 
at point 
.
is the midpoint of the minor arc 
,
intersects 
.
at 
.
.
such that 
and 
,
.
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
be fixed points on this order on a line. Let 
and 
and suppose it meets the perpendicular bisector of 
at the points 
and 
.
and 
be the other points of intersection of 
and 
with 
passes through a fixed point independent of 
Y by Adventure10
A square and an equilateral triangle are inscribed in a same circle. The seven vertices form a convex heptagon 
inscribed in the circle ( might be a hexagon if two vertices coincide). For which positions of the triangle relative to the square does have the largest and smallest area, respectively?
inscribed in the circle ( might be a hexagon if two vertices coincide). For which positions of the triangle relative to the square does have the largest and smallest area, respectively?
and three triangles. The triangles' heights from a vertex of the 7-gon are 
,
,
.
.
Then 
when ![$\alpha\in[45^{\circ},75^{\circ}]$](http://latex.artofproblemsolving.com/7/9/6/79629f8343c9b67b91016e541bcc37af0998d9e7.png)
.
or 
and maximum at 
.
and the maximum is 
where 
is the side length of the square.
