AoPS Academy
with circumcircle 
is given. Let 
be the midpoint of 
, 
be the foot of altitude from 
to 
, and 
be the intersection point of 
and 
. Point 
lies on the arc of (the one that does not contain ) such that 
. Prove that 
.
be a trapezium inscribed in a circle 
with diameter . Let be the intersection point of the diagonals and 
. The circle with center 
and radius 
meets at the points 
and 
(where is on the same side of as 
). The line perpendicular to at intersects 
at 
. Prove that 
is perpendicular to 
. in the plane of 
such that 
Let 
be the points of intersection of the medians of the triangles 
, respectively. Prove that the segments 
and 
are equal and have a common point.
in the plane of such that Let be the points of intersection of the medians of the triangles , respectively. Prove that the segments and are equal and have a common point.
and 
.
