be an acute triangle with 
,
be the projection from 
onto 
.
be a point on the extension of 
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
.
intersect 
.
.
sastifying 
such that the relation ![\[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \]](http://latex.artofproblemsolving.com/1/2/2/1229a609403904449c447225094b3d97e063efd4.png)
holds for all positive integers 
.
Y by Adventure10, Mango247, Rounak_iitr
Let 
be a triangle and let 
be a point in the interior of the triangle. The second intersection points of the lines 
and 
with the circumcircle of are 
and 
. Let 
be a point on the ray 
(these are the points on the line such that 
and lie on the same side of ). The line through parallel to 
intersects 
in 
. The line through parallel to 
intersects 
in 
. Prove that 
, and lie on a circle.
be a triangle and let 
be a point in the interior of the triangle. The second intersection points of the lines 
and 
with the circumcircle of are 
and 
. Let 
be a point on the ray 
(these are the points on the line such that 
and lie on the same side of ). The line through parallel to 
intersects 
in 
. The line through parallel to 
intersects 
in 
. Prove that 
, and lie on a circle.This post has been edited 1 time. Last edited by parmenides51, Sep 19, 2018, 12:38 PM
Reason: problem no correct
Reason: problem no correct
and 
.
.
and we conclude that 
and the claim follows. (Note that we have also proved that 
.
