AoPS Academy
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
, centroid 
and orthocenter 
. Prove that 
is constant.
opposite the vertex 
be tangent to the side 
at the point 
. Define the points 
on 
and 
on 
analogously, using the excircles opposite 
and 
, respectively. Suppose that the circumcentre of triangle 
lies on the circumcircle of triangle . Prove that triangle is right-angled.
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
and 
, Jaqueline has a ruler that allows tracing the line 
. Jaqueline also has a special object that allows the construction of a circle of diameter . Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle . Show that Jaqueline can construct the incenter of .
with circumcircle 
. Assume 
are circles which tangent internally to at 
and tangent to 
at 
, respectively. If 
are collinear, prove that 
are collinear.
with circumcircle . Assume are circles which tangent internally to at and tangent to at , respectively. If are collinear, prove that are collinear.
and tangents internally on 
and there are a chord of \Omega tangent to in 
, then 
cut the arc in its midpoint (can prove it with homothety or inversion).
, 
, 
the midpoints of the arcs 
, 
, 
. We have that , 
and 
are collinears. By Law of sines: 
and we have that 
, 
. Then 
. The similar form we have that 
and 
.
So by Ceva's Theorem we finish.
