[/quote]
,
is the foot of the altitude from 
of triangle 
and 
points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles 
.
.
![\[{x^3}{y^2}\left( {2y - x} \right) = {x^2}{y^4} - 36\]](http://latex.artofproblemsolving.com/d/5/6/d5668b2a486f5264e1cf2faeb363ac69bf91e3f0.png)
rectangles that can be covered without gaps and without overlaps with hooks such that
.
.
and 
at 
.
be an acute triangle with 
,
be the center of its circumcircle. Let 
be the reflection of 
with respect to 
.
at 
,
intersects the line through 
.
be a point on the ray 
,
.
.
be a triangle with side‐lengths 
,
,
.
be the areas of triangles 
,
,
,![\[
\frac{abc}{12R}
\;\le\;
\frac{P_1^2 + P_2^2 + P_3^2}{P}
\;\le\;
\frac{3R^3}{4\sqrt[3]{abc}}.
\]](http://latex.artofproblemsolving.com/1/3/9/139a3e15f08afb78ca0b8301f24d30db8f202207.png)
lie the following points: 
and 
on 
,
on 
,
on 
.![\[
P = AM\cap BN,\quad
R = KM\cap LN,\quad
S = KN\cap LM,
\]](http://latex.artofproblemsolving.com/7/e/a/7ea90a0f1261ed7445bbac58fb0cdd76810259f3.png)
and let the line 
meet 
.
,
are collinear.
and 
be two fixed points in the plane. For each point 
.
with 
and 
be points on segments 
and 
,
.
be the circumcircle of 
,
the circumcircle of 
.
to 
and 
,
intersect 
,
intersect 
.
is tangent to both 
.
where 
and 
are integers and 
.
intersects 
,
intersects 
.
intersects 
,
and 
intersect at 
,
and 
intersect at 
.
,
is circumcircle. Let 
.
and 
,
(opposite to 
and tangents to 
)
.
be two triangles inscribed in the same circle, centered at 
.
with respect to triangle 
.
.
lie in this order on the circle and form a convex, non-degenerate hexagon.
Y by
Let 
be the circumcircle of triangle 
, 
is the midpoint of 
and 
be the second intersection of 
and . Tangent line of at intersects at 
, let 
be a transversal of and 
be intersections of 
with . Let 
be a tangent line of . Prove that 
is cyclic
be the circumcircle of triangle 
, 
is the midpoint of 
and 
be the second intersection of 
and . Tangent line of at intersects at 
, let 
be a transversal of and 
be intersections of 
with . Let 
be a tangent line of . Prove that 
is cyclic
is harmonic 
