2006 Romanian NMO Problems/Grade 7/Problem 1
Problem
Let be a triangle and the points
and
on the sides
respectively
, such that
. Let
be a point on the line
. Prove that the lines
and
are perpendicular if and only if
is the interior angle bisector of
.
Solution
Let be a point on
such that
is the midpoint of
, then
=
, the given information is the same as \frac{LN}{BC} = \frac{AM}{AB}
ML
AC$.
Let$ (Error compiling LaTeX. Unknown error_msg)RMN
MN
NR
MN
NR
LN
NC
RLMC
CR
ML
ML
AC
A
C
R
MN
PN
NP
MC
PN
\angle MPR
PN
\angle MPC$, as requiered.