are positive real numbers, 
then
be positive real numbers with 
.
and 
Prove that
Y by Adventure10, Mango247
Let 
be an acute triangle with orthocenter 
. Segments 
and 
intersect segments 
and 
in points 
and 
respectively. The segments 
and 
meet at point 
. Let 
be the midpoint of the segment . Let 
be the reflection of the point in 
. Prove that quadrilateral 
is cyclic.
Proposed by Matko Ljulj.
be an acute triangle with orthocenter 
. Segments 
and 
intersect segments 
and 
in points 
and 
respectively. The segments 
and 
meet at point 
. Let 
be the midpoint of the segment . Let 
be the reflection of the point in 
. Prove that quadrilateral 
is cyclic.Proposed by Matko Ljulj.
and let 
,
be the midpoints of 
,
is a kite, hence 
is a diameter of nine-point circle) ![\[MN\cdot MP=MA_1^2 = MA\cdot MC.\]](http://latex.artofproblemsolving.com/a/e/1/ae1c4758f64d531f537bb994d2b7d2610a7c43e9.png)
Denoting by 
the reflection of 
is cyclic (power of 
is cyclic.
is 
-
and let 
.
-
is cyclic 
)
(nine-point circle) and ![\[ XD\cdot XN = XA_1\cdot XM = XC_1\cdot XA_1 = XA\cdot XC.\]](http://latex.artofproblemsolving.com/7/2/2/7226c365fe482775c02f74a7ad03500ab74a5f01.png)
.
is the bisector of 
and 
is cyclic (and all those points lie onto the 9-point circle of 
)
.
and 
is symmetrical of 
w.r.t. 
intersect circle 
at 
and 
,
becomes 
,
w.r.t. 
we have 
;
,
.
are the polars of 
with respect to the circumcircle of 
.
and 
are similar thus
.
so 
.
.
,
is the orthocenter of 
,
.
be Cyclic.
as acute angles with perpendicular sides and since 
due to the isosceles triangle 
, 
and then its obvious.
are autopolar by Brocard's and 
We can observe that 
