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be an acute triangle with incentre 
and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and 
such that 
and 
are parallelograms (with 
, and 
). Let 
be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
satisfies the equation![\[
(AB + CD)^2 + (AD + BC)^2 = (AC + BD)^2,
\]](http://latex.artofproblemsolving.com/7/d/a/7daf2d6ac00609b5252b2b8b05d96a5ca0952a94.png)
then must be a rectangle.
, let the incircle 
touch sides 
at 
, respectively. Let 
lie on the line through 
perpendicular to 
. Let 
be the intersections of 
with 
, respectively. Let 
be the projections of 
onto line . Let 
be the second intersections of 
with the incircle . Let be the intersection of 
and 
. Prove that the line 
bisects segment 
.
be a triangle with circumcircle 
, 
-angle bisector 
, and -median 
. Suppose that meets 
at and meets again at 
. A line 
parallel to meets , at 
, 
respectively, so that is between and . The circle with diameter 
meets again at 
. varies, show that 
is constant.
. Let and 
be the feet of the altitudes from to and from 
to 
respectively.
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by the second intersection of line 
with , and by 
the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
, prove that![\[ a = b\cos C + c\cos B. \]](http://latex.artofproblemsolving.com/7/e/d/7ed699f18ede191e5b89bc79d7f0592294347818.png)
be a rectangle and any point. Let 
be the foots of the perpendiculars from to the lines 
, respectively. Let the perpendicular bisectors of 
and 
intersect at , and those of 
and 
intersect at . Prove that the lines and 
are parallel., let be the midpoint of , and the midpoint of 
. Suppose 
. Prove that![\[
2\angle AED-\angle DCB =180^\circ.
\]](http://latex.artofproblemsolving.com/b/d/0/bd01635e002970e841cef9dcb98b154ad6a7a202.png)
, define![\[
\alpha = |\angle ADB - \angle ACB| = |\angle DAC - \angle DBC|
\quad\text{and}\quad
\beta = |\angle ABD - \angle ACD| = |\angle BAC - \angle BDC|.
\]](http://latex.artofproblemsolving.com/3/e/e/3eef0425824a79ba8615288dcfe2df79fca048b1.png)
Prove that![\[
AC \cdot BD = AD \cdot BC \cos\alpha + AB \cdot CD \cos\beta.
\]](http://latex.artofproblemsolving.com/d/4/a/d4adc5d510093da3520b26cbce172cf791b2847a.png)
, let 
be the sum of two opposite angles. Prove that![\[
AC^2 \cdot BD^2 = AB^2 \cdot CD^2 + AD^2 \cdot BC^2 - 2AB \cdot CD \cdot AD \cdot BC \cos\theta.
\]](http://latex.artofproblemsolving.com/4/2/4/424dfbb285f46ccc5524b439028bd2ef54569c03.png)
be the point of intersection of 
and 
. We can easily see that is the radical axis of the two circles. Thus, since lies on , 
. Hence, 
is a cyclic quadrilateral, so 
, or 
. Therefore, 
is a cyclic quadrilateral 
. This means lies on , so , and are concurrent.
