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such that for all 
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be an acute triangle with 
and orthocenter 
. Let 
and 
be the feet of the altitudes from 
and 
onto sides 
and 
, respectively. Let 
be the midpoint of 
, and 
be the midpoint of 
. Let the perpendicular line through to 
meet at 
and at 
. The line 
meets at 
and at 
. Let 
be the second intersection point (different from ) of the circumcircles of triangles 
and 
. Prove that the points , , , , and lie on the same circle.
with a right angle , the angle bisector 
and the median 
are drawn, they intersect at point 
. Line 
intersects side at point 
. Find the area of triangle 
if it is known that 
.
with a right angle , the angle bisector and the median are drawn, they intersect at point . Line intersects side at point . Find the area of triangle if it is known that .
and 
Prove that
and 
Prove that