be a triangle with 
let 
be the circumcircle of 
and let 
be its radius. Point 
such taht 
and point 
is the foot of the perpendicular from 
.
mets 
.
is chosen on line 
such that 
and 
lie on a line in that order. Finally, let 
be a point satisfying 
and 
.
,
has two real roots, so that the one is the square of the other.
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
.
.
such that: ![\[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]](http://latex.artofproblemsolving.com/e/c/f/ecf7fb3c2805290f2cb07b7fdd7c73e1c7a55b99.png)
Y by Adventure10, Mango247
Let 
be the centre of the incircle of 
. Points 
are the intersection points of the circles circumscribed about triangles 
respectively with the bisectors of the angles at 
respectively 
. Also 
is the midpoint of segment 
, 
is the reflection of with respect to and 
is the reflection of with respect to line . Prove that the points 
and lie on the same circle.
be the centre of the incircle of 
. Points 
are the intersection points of the circles circumscribed about triangles 
respectively with the bisectors of the angles at 
respectively 
. Also 
is the midpoint of segment 
, 
is the reflection of with respect to and 
is the reflection of with respect to line . Prove that the points 
and lie on the same circle.This post has been edited 1 time. Last edited by StefanS, Apr 13, 2012, 5:21 PM
is on the segment 
is a deltoid
)
.
,
is a parallelogram (their diagonals divide in half), thus 
.
