,![\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
\]](http://latex.artofproblemsolving.com/3/2/b/32b41635c159921e11245369e66185ec69b0d785.png)
,
.
is the midpoint of 
is the midpoint of 
.
to the circumcircle of 
.
passes through one of the intersections of circles 
with 
.
are positiv real number, such that 
,
Y by Mango247, Mango247, Mango247
Let 
be a non-right triangle such that 
. We denote by 
the projection of 
on 
, and 
the projections of on 
respectively . Line 
i ntersects at point 
. We consider the semicircle of diameter ![$[CD]$](//latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
located in the same semiplane by 
that contains . Let 
be the point of this semicircle whose projection on 
is 
. Show that 
is tangent to the circumcircle of 
.
be a non-right triangle such that 
. We denote by 
the projection of 
on 
, and 
the projections of on 
respectively . Line 
i ntersects at point 
. We consider the semicircle of diameter ![$[CD]$](http://latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
located in the same semiplane by 
that contains . Let 
be the point of this semicircle whose projection on 
is 
. Show that 
is tangent to the circumcircle of 
.This post has been edited 1 time. Last edited by parmenides51, Sep 27, 2020, 10:28 PM
,
is cyclic.
,
is cyclic.
.
.
,
is tangent to the circumcircle of 
.
