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be a triangle with 
. The tangent to the circumcircle of 
at 
intersects 
at 
. The angle bisector of 
intersect at 
. Suppose that the perpendicular bisector of 
intersect 
at 
, respectively. Show that 
lie the following points: 
and 
on 
, 
on 
, and 
on 
. Let![\[
P = AM\cap BN,\quad
R = KM\cap LN,\quad
S = KN\cap LM,
\]](http://latex.artofproblemsolving.com/7/e/a/7ea90a0f1261ed7445bbac58fb0cdd76810259f3.png)
and let the line 
meet at 
. Prove that the points 
, , and 
are collinear.
be the incenter of triangle . Let 
, 
, and 
be the orthocenters of triangles 
, 
, and 
, respectively. Prove that the lines through , , and , parallel to 
, 
, and 
, respectively, are concurrent.
, points 
and 
are isogonal conjugates. The tangent to 
at and the tangent to 
at Q, meet at 
. 
intersects 
at . Prove that points ,, and are concyclic.
and 
be the incenter and the circumcenter of a triangle , respectively, and let
be the exterior bisector of angle . The line through perpendicular to 
meets the and at points and , respectively. Prove that 
. satisfy the equality, 
prove that the triangle must have a right angle.
be a positive integer. Show that there exist a polynomial 
with integer coefficient that satisfy the following is at most 
for each 
such that 
exist in the first quadrant of the 
plane for right 
. If hypothenuse 
is represented by the line 
, then let side of similar right 
be represented by 
such that 
. We are given that 
, which satisfies:
. Next, let:
(i);
(ii)
;
;
;
(iii).
.
. Prove that
W here 
and 
Show that
.
.![\[\sum ab +\sum ab^2 \le 6\]](http://latex.artofproblemsolving.com/9/f/6/9f632ab1e6df5029bcd4416eb06623123ad2cd8c.png)
and 
be two similar right-angled triangles (in the same plane) with 
such that 
lie on opposite sides of the line 
.
,
.