by ![\[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\]](http://latex.artofproblemsolving.com/c/b/d/cbd9b412893774283a438296cdeb05b80cc878bf.png)
Let 
and 
be two real numbers such that 
.
and 
by 
,
,
for 
.
such that ![\[(a_n - a_{n-1})(b_n-b_{n-1})<0.\]](http://latex.artofproblemsolving.com/a/5/c/a5c8844d595bb1186d4804beadd69a3bc1d40ea3.png)
is marked on side 
,
- point 
so that the circumscribed circle of 
is tangent to line segment 
.
is tangent to line 
.
and circumcentre 
.
cross 
.
,
cross 
at 
cross 
at 
.
and 
are concurrent on the external bisector of 
.
Y by Adventure10, Mango247
Let 
be the centroid of a triangle 
, and 
be the midpoint of 
. Let 
be on 
and 
on 
such that the points , , and are collinear and 
and are parallel. Suppose that 
and 
intersect at 
and 
and 
intersect at 
. Show that triangle 
is similar to triangle .
be the centroid of a triangle 
, and 
be the midpoint of 
. Let 
be on 
and 
on 
such that the points , , and are collinear and 
and are parallel. Suppose that 
and 
intersect at 
and 
and 
intersect at 
. Show that triangle 
is similar to triangle .
,
,
.
,
,
.
)
and 
with sides 
and 
respectively. By Menelaos in CC'X and line GB we have 
and by Thales 
.
Hence 
,
and 
and 
.
and 
.
.
by the midpoint of 
.
.
it results that 
.
.
be the midpoint of 
.
.
.
and 
have parallel sides. So they're similar.
and 
:
we know that :
by menlaues on triangle 
and point 
and also 
and point 
so we know that:
be the medial triangle WRT 
and then apply Ceva on 
,
and 
to obtain 
.
.
where 
is the displacement vector, so its easy to see 
.
with similarity factor 
.
.
meet 
center 
takes 
and 
and 
,
.
.
= 
.
+ 
,
and so 
.
and 
are similar, so 
is parallel to 
.
is parallel to 
its length, so 
is parallel to 
is parallel to 
is similar to 
.
,
.
and 
,
and 
are homothetic, as are triangles 
and 
;
.
,
with 
.
,
which implies 
.
,
