such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
be the orthogonal projection of 
on 
and 
are the midpoints of 
and 
respectively. Let 
and 
are points on the minor arcs 
and 
of the circumcircle of triangle 
.
and 
are tangent if and only if 
.
and 
on 
and 
.
with 
and 
are 
and 
with 
and 
are 
,
and 
are concurrent.
,
,
and circumcentre 
.
is a parallellogram. Suppose that there exists a point 
on 
to 
.
be the midpoint of 
.
,
.
Y by Adventure10, Mango247
Triangle 
has an incenter 
l its incircle touches the side 
at 
. The line through parallel to 
meets the incircle again at 
and the tangent to the incircle at meets 
at points 
respectively. Prove that triangle 
is similar to triangle .
has an incenter 
l its incircle touches the side 
at 
. The line through parallel to 
meets the incircle again at 
and the tangent to the incircle at meets 
at points 
respectively. Prove that triangle 
is similar to triangle .This post has been edited 2 times. Last edited by nsato, Oct 9, 2019, 12:18 AM
.
is 
.
.
is 
Let 
be the points where 
touches 
respectively. Let 
and let the midpoints of 
be 
respectively. Obviously 
wrt 
and it is given that 
Since the polar of 
)
so we obtain 
Therefore the quadrilateral 
apart from being a parallelogram, also has 
and 
,
Thus 
is a rectangle, and therefore cyclic. Note that an inversion about 
)
to 
to 
to 
and 
to 
Therefore the new quadrilateral 
is also cyclic. So, 
and therefore 
.
is the bisector of 
,
is the bisector of 
,
,
