= 2∠AMP.
,
not dividing any of them, and an integer 
such that if one considers the infinite sequence 
and looks at the highest power of 
(which may depend on 
)
.
be a positive integer. Consider 
points on the circumference of a circle.
}
and 
in this set at last in one of the permutation
Y by Adventure10, Mango247
Let 
be a convex quadrilateral such that the line 
bisects the angle 
The circumcircle of triangle 
intersects the sides 
and 
in the points 
and 
respectively. The line through 
and parallel to 
intersects the lines 
and 
at the points 
and 
respectively. Prove that the points 
and 
lie on a common circle.
be a convex quadrilateral such that the line 
bisects the angle 
The circumcircle of triangle 
intersects the sides 
and 
in the points 
and 
respectively. The line through 
and parallel to 
intersects the lines 
and 
at the points 
and 
respectively. Prove that the points 
and 
lie on a common circle.
,
are concyclic .
are concyclic .
,
are concyclic .
is tangent to 
at 
.
which also lies on the tangent to 
.
,
collinear. Now 
,
be the points on 
respectively, so that 
.
intersects 
and 
in the points 
and 
lie on a common circle.
the claim follows.
to 
is also tangent to 
.
as 
.
.
.![[asy]
size(7cm);defaultpen(fontsize(10pt));pen org=magenta;pen med=mediummagenta;pen light=pink;pen deep=deepmagenta;pen dark=darkmagenta;pen heavy=heavymagenta;
pair O,A,B,C,D,K,L,P,Q,R;
O=(0,0);A=dir(120);B=dir(200);C=dir(340);path w=circumcircle(A,B,C);D=extension(incenter(A,B,C),A,B,C);K=0.45*B+0.55*A;
L=extension(A,C,K,foot(K,O,midpoint(B--C)));path g=circumcircle(A,K,L);P=intersectionpoints(K--D,g)[1];Q=intersectionpoints(L--D,g)[1];R=extension(B,C,P,Q);
draw(A--B--C--cycle,deep);draw(w,deep);draw(g,deep);draw(D--K,med);draw(D--L,med);draw(R--A,deep);draw(R--Q,light+dashed);draw(R--B,deep);
dot("$A$",A,dir(A));
dot("$B$",B,dir(B));
dot("$C$",C,dir(C));
dot("$D$",D,dir(D));dot("$K$",K,dir(K));dot("$L$",L,dir(L));dot("$Q$",Q,S);dot("$P$",P,dir(P));dot("$R$",R,dir(R));
[/asy]](http://latex.artofproblemsolving.com/e/2/d/e2d6fb0a2f033a1fc82f13d939b2582f82b3f85f.png)
at the vertex 
be the intersection of line 
and
,
be the circle passes through 
is cyclic.
,
are tangent. Since 
and 
,
is an isosceles trapezoid, and we are done.
