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 L be a point on BC such that N is the midpoint of LC, then 2CN=LC, the given information is the same as ''LC/BC=AM/AB'', applicating Thales theorem it follows that ML is parallel to AC. |
| − | + | ||
| + | Let R be the point on MN such that MN=NR, in view of MN=NR and LN=NC it follows that RLMC is a parallelogram, implying that CR is parallel to ML, but we know that ML is parallel to AC, then A,C,R are collineal. | ||
| + | |||
| + | 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. | ||
==See also== | ==See also== | ||
Revision as of 19:12, 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 
.
Let L be a point on BC such that N is the midpoint of LC, then 2CN=LC, the given information is the same as LC/BC=AM/AB, applicating Thales theorem it follows that ML is parallel to AC.
Let R be the point on MN such that MN=NR, in view of MN=NR and LN=NC it follows that RLMC is a parallelogram, implying that CR is parallel to ML, but we know that ML is parallel to AC, then A,C,R are collineal.
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.
