Difference between revisions of "2006 Romanian NMO Problems/Grade 7/Problem 1"
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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>. | ||
==Solution== | ==Solution== | ||
+ | {{solution}} | ||
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==See also== | ==See also== | ||
*[[2006 Romanian NMO Problems]] | *[[2006 Romanian NMO Problems]] | ||
[[Category:Olympiad Geometry Problems]] | [[Category:Olympiad Geometry Problems]] |
Revision as of 07:33, 27 August 2008
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
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