Difference between revisions of "2006 Romanian NMO Problems/Grade 7/Problem 1"
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==Problem== | ==Problem== | ||
Let <math>ABC</math> be a triangle and the points <math>M</math> and <math>N</math> on the sides <math>AB</math> respectively <math>BC</math>, such that <math>2 \cdot \frac{CN}{BC} = \frac{AM}{AB}</math>. Let <math>P</math> be a point on the line <math>AC</math>. Prove that the lines <math>MN</math> and <math>NP</math> are perpendicular if and only if <math>PN</math> is the interior angle bisector of <math>\angle MPC</math>. | Let <math>ABC</math> be a triangle and the points <math>M</math> and <math>N</math> on the sides <math>AB</math> respectively <math>BC</math>, such that <math>2 \cdot \frac{CN}{BC} = \frac{AM}{AB}</math>. Let <math>P</math> be a point on the line <math>AC</math>. Prove that the lines <math>MN</math> and <math>NP</math> are perpendicular if and only if <math>PN</math> is the interior angle bisector of <math>\angle MPC</math>. | ||
− | |||
− | Let | + | ==Solution== |
+ | Let <math>L</math> be a point on <math>BC</math> such that <math>N</math> is the midpoint of <math>LC</math>, then <math>2CN</math>=<math>LC</math>, the given information is the same as \frac{LN}{BC} = \frac{AM}{AB}<math>, applicating Thales theorem it follows that </math>ML<math> is parallel to </math>AC<math>. | ||
− | MN is perpendicular to PN if and only if NP is the perpendicular bisector of MC if and only if PN is the angle bisector of MPR if and only if PN is the angle bisector of MPC, as requiered. | + | Let </math>R<math> be the point on </math>MN<math> such that </math>MN<math>=</math>NR<math>, in view of </math>MN<math>=</math>NR<math> and </math>LN<math>=</math>NC<math> it follows that </math>RLMC<math> is a parallelogram, implying that </math>CR<math> is parallel to </math>ML<math>, but we know that </math>ML<math> is parallel to </math>AC<math>, then </math>A<math>,</math>C<math>,</math>R<math> are collineal. |
+ | |||
+ | </math>MN<math> is perpendicular to </math>PN<math> if and only if </math>NP<math> is the perpendicular bisector of </math>MC<math> if and only if </math>PN<math> is the angle bisector of </math>\angle MPR<math> if and only if </math>PN<math> is the angle bisector of </math>\angle MPC$, as requiered. | ||
==See also== | ==See also== |
Revision as of 19:18, 11 October 2013
Problem
Let be a triangle and the points
and
on the sides
respectively
, such that
. Let
be a point on the line
. Prove that the lines
and
are perpendicular if and only if
is the interior angle bisector of
.
Solution
Let be a point on
such that
is the midpoint of
, then
=
, the given information is the same as \frac{LN}{BC} = \frac{AM}{AB}
ML
AC$.
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.