[/quote]
,
and 
.
Equality holds when 
or 
Equality holds when 
intersect its circumcircle 
at 
.
determine a triangle 
,
is relative to 
and so on. Prove that lines 
are concurrent at a point 
and that 
and 
.
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
at 
,
intersect 
at 
.
and 
are either parallel or coincide.
and 
.
Where 
such that :
.
be reals such that 
Prove that 
Let 
Prove that 
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using 
colors, such that among any 
consecutive beads every color appears at least once. Find the largest value of 
for which this task is 
possible.
This easy via the binomial theorem for the quantity is just 
,
satisfies 
.
,
.
and a constant 
,
and 
are also friendly polynomials. Prove that 
.
Y by Adventure10, Mango247, and 1 other user
Let be given a tetrahedron whose circumcenter is 
. Draw diameters 
of the circumsphere of 
. Let 
be the centroids of triangle 
. Prove that 
are concurrent at a point, say, 
. Prove that the line through and a midpoint of a side of is perpendicular to the opposite side.
. Draw diameters 
of the circumsphere of 
. Let 
be the centroids of triangle 
. Prove that 
are concurrent at a point, say, 
. Prove that the line through and a midpoint of a side of is perpendicular to the opposite side.
is the centroid of the tetrahedron 
Let 
By Menelaus' theorem for 
cut by 
keeping in mind that 
and 
we get 
i.e. 
Likewise, 
and 
pass through 
be the midpoints of the edges 
Since 
is the midpoint of the bimedian 
then it follows that 
is a parallelogram with diagonal intersection 
i.e. skew lines 
and 
are orthogonal. Analogously, the lines joining the midpoints of 
with 
respectively.
