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be an equilateral triangle with circumcenter 
. Let 
and 
be two points on segments 
and 
, respectively, such that 
. If 
is the reflection of with respect to line 
, prove that lines 
and 
are parallel.
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
, there exists a non-degenerate triangle with sides of lengths![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
be an acute-angled triangle with 
. Let 
be the circumcircle of . Let 
be the midpoint of the arc 
of containing 
. The perpendicular from to 
meets 
at 
and meets again at 
. The line through parallel to meets line 
at 
. Denote the circumcircle of triangle 
by 
. Let meet again at 
. Prove that the line tangent to at 
meets line on the internal angle bisector of 
.
be a cyclic pentagon inscribed in a circle of centre 
which has angles 
. Show that the diagonals 
and 
meet at a point belonging to the diameter 
.
as the perpendicular bisector of 
thus we prove that the circle tangent to 
through 
touches the circle 
cut again at 
is P-isosceles 
is perpendicular bisector of 
cutting the arc 
of 
at its midpoint 
Since 
then 
C-excenter of 
is on 
By Sawayama's lemma, we conclude that the circle tangent to through is a Thebault circle of the cevian 
of externally tangent to its circumcircle 
.
and 
.
bisect segment 
,
is incenter of 