Y by tiendung2006, ImSh95, GioOrnikapa, nervy, Siddharth03, Miku_
Let 
be a triangle with circumcenter 
. 
are midpoints of 
. 
intersects 
at 
. Let 
be the circumcenter of 
and 
. Prove that 
and tangent line of 
at 
are concurrent
be a triangle with circumcenter 
. 
are midpoints of 
. 
intersects 
at 
. Let 
be the circumcenter of 
and 
. Prove that 
and tangent line of 
at 
are concurrent
at 
cut each others at 
.
and 
are concurrent.
-
(below marked as 
)![[asy]
unitsize(80);
pair A,B,C,O,X,Y,K,E,F,N;
A=dir(110); B=dir(215); C=dir(-35); O = circumcenter(A,B,C); X = midpoint(A--B); Y = midpoint(A--C); K = extension(A,O,B,C); F = circumcenter(A,X,K); E = circumcenter(A,Y,K); N = extension(Y,F,X,E);
draw(A--B--C--cycle);
draw(X--K); draw(A--K);
draw(A--K); draw(K--Y);
draw(circumcircle(A,X,K), red);
draw(circumcircle(A,Y,K), red);
draw(circumcircle(A,B,C), red);
draw(circumcircle(A,N,K), red + linetype("4 4"));
draw(Y--N); draw(A--N);
draw(N--C); draw(E--N);
dot("$O$", O, NW * dir(10));
dot("$A$", A, N * dir(40));
dot("$B$", B, SW);
dot("$C$", C, SE);
dot("$X$", X, NW);
dot("$Y$", Y, E * dir(-40));
dot("$K$", K, SE);
dot("$E$", E, E);
dot("$F$", F, NW);
dot("$N$", N, NE);
[/asy]](http://latex.artofproblemsolving.com/c/5/f/c5fb0af000d91270f76df5ee9193c39a6ae2af4f.png)
be a triangle with circumcentre 
.
are the midpoints of 
.
in 
such that 
.
intersect 
at 
.
.
intersect 
at 
.
is a point such that 
,
intersects 
at 
.
.
intersects 
at 
are the reflection of 
about 
.
intersects 
at 
but 
is the radical axis of 2 circles. So we have the new problem
