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, 
and 
are the midpoints of sides 
and 
respectively. Point 
lies inside the triangle. Let 
. Prove that 
.
, 
, and 
of a hexagon 
inscribed in a circle 
intersect at a point 
, and the acute angle between any two of them is 
. Let 
be the radius of the circle tangent to segments 
and 
and internally tangent to ; the radii 
, 
, 
, 
, and 
are defined similarly. Prove that![\[
r_{AB}r_{CD} + r_{CD}r_{EF} + r_{EF}r_{AB} = r_{BC}r_{DE} + r_{DE}r_{FA} + r_{FA}r_{BC}.
\]](http://latex.artofproblemsolving.com/0/d/6/0d66d6685bbc97c7553e9ed3ba14ea0dc3328b49.png)
, let 
be the circumcentre, 
be the orthocentre, and 
be the centroid.
satisfy the following condition:
is constant.
. Find 
.
be the orthocenter of acute-angled triangle 
. M is the midpoint of side BC. P is on the median AM such that 
. Prove that P lies on the circumcircle 
of triangle BHC.
.
, hence BMPE and CMPD are two cyclic quadrilateral.
Circle 
with diameter 
and circumcircle of triangle 
intersect at point 
Prove that the straight line 
pass through the midpoint of 
