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, Mango247, and 1 other user
Let 
be a tetragonal.
(a) If the plane
cuts 
find the necessary and sufficient condition such that the area formed from the intersection of the plane and the tetragonal be a parallelogram. Prove that the problem has three solutions in this case.
(b) Consider one of the solutions of (a). Find the situation of the plane for which the parallelogram has maximum area.
(c) Find a plane for which the parallelogram be a lozenge and then find the length side of his lozenge in terms of the length of the edges of 
be a tetragonal.(a) If the plane
cuts 
find the necessary and sufficient condition such that the area formed from the intersection of the plane and the tetragonal be a parallelogram. Prove that the problem has three solutions in this case.(b) Consider one of the solutions of (a). Find the situation of the plane
for which the parallelogram has maximum area.(c) Find a plane
for which the parallelogram be a lozenge and then find the length side of his lozenge in terms of the length of the edges of 
