touch sides 
at 
,
perpendicular to 
.
be the intersections of 
with 
,
be the projections of 
onto line 
be the second intersections of 
with the incircle 
be the intersection of 
and 
.
bisects segment 
.
be the incentre of 
such that the incircle touches the sides 
and 
respectively. 
and 
meet the incircle at 
respectively. 
and 
intersect at 
Prove that 
bisects 
then prove 
![$P\in\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/f/3/d/f3d763f4197792a2956273485eb1e94700a43044.png)
has degree 
having 
.
is the leading coefficient of 
Y by Davi-8191, amar_04, Adventure10
Let 
be a triangle, and let 
be a distinct point on the plane. Moreover, let 
be a homothety of with ratio 
and center , and let 
and 
be the circumcenters of and , respectively. The circumcircles of 
, 
, and 
meet at points 
, 
, and 
, different from 
, 
, and 
. In a similar way, the circumcircles of 
, 
, and 
meet at 
, 
, and 
, different from 
, 
, 
. Let 
and 
be the circumcenters of 
and 
, respectively. Prove that 
is parallel to 
.
Proposed by Mateus Thimóteo, Brazil.
be a triangle, and let 
be a distinct point on the plane. Moreover, let 
be a homothety of with ratio 
and center , and let 
and 
be the circumcenters of and , respectively. The circumcircles of 
, 
, and 
meet at points 
, 
, and 
, different from 
, 
, and 
. In a similar way, the circumcircles of 
, 
, and 
meet at 
, 
, and 
, different from 
, 
, 
. Let 
and 
be the circumcenters of 
and 
, respectively. Prove that 
is parallel to 
.Proposed by Mateus Thimóteo, Brazil.
This post has been edited 2 times. Last edited by Kowalks, Jul 22, 2017, 5:02 PM
![[asy]
size(350);
defaultpen(fontsize(10pt));
pair P, A, B, C, AA, BB, CC, X, Y, Z, XX, YY, ZZ, W, WW, O, OO;
A = dir(120);
B = dir(220);
C = dir(340);
P = (-3,0);
AA = 2*A-P;
BB = 2*B-P;
CC = 2*C-P;
O = circumcenter(A, B, C);
OO = circumcenter(AA, BB, CC);
draw(A--B--C--cycle, orange+linewidth(1.2));
draw(AA--BB--CC--cycle, orange+linewidth(1.2));
draw(P--AA^^P--BB^^P--CC, red);
path w1, w2, w3, g1, g2, g3;
w1 = circumcircle(A, BB, CC);
w2 = circumcircle(AA, B, CC);
w3 = circumcircle(AA, BB, C);
g1 = circumcircle(AA, B, C);
g2 = circumcircle(A, BB, C);
g3 = circumcircle(A, B, CC);
draw(w1^^w2^^w3, heavygreen+dashed);
draw(g1^^g2^^g3, blue+dashed);
X = IP(w2, w3, 0);
Y = IP(w1, w3, 1);
Z = IP(w1, w2, 0);
XX = IP(g2, g3, 1);
YY = IP(g1, g3, 0);
ZZ = IP(g1, g2, 0);
W = circumcenter(X, Y, Z);
WW = circumcenter(XX, YY, ZZ);
draw(O--W, magenta);
draw(OO--WW, magenta);
draw(X--Y--Z--cycle, purple);
draw(XX--YY--ZZ--cycle, purple);
clip(Circle((O+W)/2, 4.5));
dot("$A$", A, dir(A));
dot("$B$", B, dir(B));
dot("$C$", C, dir(270));
dot("$A'$", AA, dir(70));
dot("$B'$", BB, dir(215));
dot("$C'$", CC, dir(CC));
dot("$P$", P, dir(P));
dot("$X$", X, dir(30));
dot("$Y$", Y, dir(315));
dot("$Z$", Z, dir(270));
dot("$X'$", XX, dir(300));
dot("$Y'$", YY, dir(YY));
dot("$Z'$", ZZ, dir(140));
dot("$O$", O, dir(90));
dot("$O'$", OO, dir(45));
dot("$W$", W, dir(270));
dot("$W'$", WW, dir(180));
[/asy]](http://latex.artofproblemsolving.com/7/9/5/7951f5025f9a294f92d7989b6eb5222853dbdb98.png)
