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such that for all 
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be an acute scalene triangle , 
be the orthogonal projection of 
on 
, 
and 
are the midpoints of 
and 
respectively. Let 
and 
are points on the minor arcs 
and 
of the circumcircle of triangle respectively such that 
. Show that the circumcircles of triangles 
and 
are tangent if and only if lies on 
.
+1 w be a triangle and the points 
and 
on , and on , and and on be such that 
and 
. The intersections of 
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are and 
, respectively. Prove that the lines 
and 
are concurrent.
be an acute-angled triangle with 
, orthocenter 
, circumcircle 
and circumcentre 
. Let be the midpoint of and let be a point such that 
is a parallellogram. Suppose that there exists a point 
on and on the opposite side of 
to such that 
. Let 
be the midpoint of 
. Prove that if 
, then 
.
that passes through the centroids of each face of a given tetrahedron inscribed in a unit sphere with center . Also, determine the distance from to the center of as a function of the edges of 
be the circumsphere of the tetrahedron 
and 
be the circumsphere of the tetrahedron whose vertices are the centroids 
of its faces against 
Segments 
concur at the centroid 
of 
such that divides them in the same ratio 
Tetrahedra 
and are homothetic through the homothety with center and coefficient 
Therefore, 
and 
Let 
be the edges of 
By Leibniz theorem for the circumcenter of we have
