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be three collinear points and 
three other collinear points. Let 
be the intersection of the lines 
and 
,respectively. If 
and 
.Prove that 
. Let 
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of 
.
be a fixed acute-angled triangle. Consider some points 
and 
lying on the sides 
and 
, respectively, and let 
be the midpoint of 
. Let the perpendicular bisector of intersect the line 
at 
, and let the perpendicular bisector of 
intersect the lines and at 
and 
, respectively. We call the pair 
, if the quadrilateral 
is cyclic.
and 
are interesting. Prove that 
is given. Over the side draw an equilateral triangle 
on the outside. The midpoint of the segment 
is 
and the midpoint of the side 
is 
. Prove that 
. be an acute triangle. Let 
be the reflection of point over the line . Let 
and be the circumcenter and the orthocenter of triangle , respectively, and let be the midpoint of segment 
. Let 
and 
be the points where the reflection of line 
with respect to line 
intersects the circumcircle of triangle , where point lies on the arc not containing . If 
is a point on the line 
such that 
, prove that 
.
be a regular hexagon with sidelength s. The points 
and 
are on the diagonals 
and 
, respectively, such that 
. Prove that the three points 
, and are on a line.
, and 
be the angles of a triangle, and let 
, and 
. If 
, show that![\[s(s - a)(s - b)(s -c) \geq 0.\]](http://latex.artofproblemsolving.com/1/3/4/1344a0cbbd4e1aa4bcf6a1e777501c62daff32b3.png)
has 
. Point is the foot of the perpendicular from to . Points and lie on sides and 
, respectively, such that lies inside triangle 
and ![\[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \]](http://latex.artofproblemsolving.com/f/8/c/f8c35912c531eb2dae8f62b74666b7757d2b9ba3.png)
Prove that line is tangent to the circumcircle of triangle 
.
cannot cover entirely a square surface of side 
, but they can cover more than 
of it.
is chosen inside a convex quadrilateral . Could it happen that
![[asy]
size(150);
draw((-1,0)--(3,0)); // ground line
draw(circle((0,1),1)); // left big circle
draw(circle((2,1),1)); // right big circle
draw(circle((1,0.25),0.25)); // small circle in between
[/asy]](http://latex.artofproblemsolving.com/d/0/d/d0d9cb2706e9b913e2051e224b8e64537c7b8332.png)
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