be the midpoint of the side 
.
intersects the side 
.
intersect the lines 
and 
respectively. If the midpoint of 
is 
,
.
for which 
is a perfect square.
,
lie on a line in this order and satisfy 
.
and 
,
is acute, and let 
lie on lines 
and 
,
and 
and 
and 
Y by Davi-8191, Adventure10, Mango247
Let 
be a scalene triangle and 
, 
and 
be points on the lines 
, 
and 
, respectively, such that 
. The circumcircles of 
and 
intersect at 
. Prove that is on the circumference which diameter has ends in the ortocenter 
and in the baricenter 
of .
be a scalene triangle and 
, 
and 
be points on the lines 
, 
and 
, respectively, such that 
. The circumcircles of 
and 
intersect at 
. Prove that is on the circumference which diameter has ends in the ortocenter 
and in the baricenter 
of .
to be 
.
,
,
to be the intersection of 
and 
.
,
respectively. Let 
.
is on 
and ![\[
\angle EY_AX_A=\angle BCX_a=\angle X_AAB=\angle X_AAE.
\]](http://latex.artofproblemsolving.com/e/7/d/e7d5bc4f7bba0080b5d6808e7a38654d96f7d874.png)
Of course by Miquel's theorem 
.
with respect to 
,
,
,
.
always passes through 
.
and 
are projective, so we only need to check 3 cases. When 
,
,
follows similarly, we have the desired result.
is projective, as well as 
.
and 
intersect on 
,
,
,
,
.
.
and similarly for the others. Thus 
and similarly for the other points. Thus there is a spiral similarity at 
to 
so 
since the radius of 
is 
.
we obtain 
,
.
of 
;
using 
.
,
HM point lies on 
,
.
are collinear along with 
implies 
.
lies on the circle with diameter 
are concurrent at 
and we are done.
