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be fixed points on this order on a line. Let 
be a variable circle through 
and 
and suppose it meets the perpendicular bisector of 
at the points 
and 
. Let 
and 
be the other points of intersection of 
and 
with . Prove that 
passes through a fixed point independent of .
be a convex quadrilateral with 
. Let 
be the point of intersection of 
with 
and let 
be the midpoint of 
. On the extension of , beyond the point , we pick a point such that 
. Let 
and 
be the projections of 
and respectively on 
. The circumcircle of the triangle 
meets again at the point 
. If 
is the point of intersection of with , prove that the lines 
and 
intersect on the line 
.
. Let 
be the circumcenter of this triangle . Let 
, 
, be the feet of the altitudes of triangle from the vertices , 
, , respectively. Denote by 
, 
, 
the midpoints of these altitudes 
, 
, 
, respectively. The incircle of triangle has center and touches the sides , 
, 
at the points , , 
, respectively. Prove that the four lines 
, 
, 
and 
are concurrent. (When the point concides with , we consider the line as an arbitrary line passing through .)
be an acute triangle and let 
be an interior point for it such that 
. Prove that
When does equality hold?
, 
, and 
are the bisectors of the interior angles of triangle , with , , and lying on the perimeter. If angle 
is 
degrees, determine all possible values of angle 
.
and intersect at point and let and intersect at .Consider quadirateral 
.Then we get that 
.Thus the condition 
implies 
=
.Hence F is excenter of the triangle 
.Thus we get 
=
=
, where is intersection of and 
.Then we get 
