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 .
be a triangle with incenter 
and let 
be the midpoints of sides 
, respectively. Lines 
and 
meet at 
and lines 
and 
meet at . Line meets sides and 
at 
and 
, respectively. Prove that 
be a triangle with incenter and let be the midpoints of sides , respectively. Lines and meet at and lines and meet at . Line meets sides and at and , respectively. Prove that 
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with midpoints 
of sides 
respectively and it’s incenter 
. Let 
be points of tangency of incircle with sides 
. Then lines 
are concurrent.
and analogously 
. Thus 
and we’re done by tangent segments theorem: 
. QED
, so 
. Thus, 
(and by the same argument, these all also equal 
), so 
is isoceles.
, so 
.
.
.
,
,
and we are done.
and
Observe
Because 
we have similarity in the same orientation of isosceles triangles
Since 
the equality proves