# 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>. | ||

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− | Let | + | ==Solution== |

+ | Let <math>L</math> be a point on <math>BC</math> such that <math>N</math> is the midpoint of LC, then <math>2CN</math>=<math>LC</math>, the given information is the same as <math>\frac{LC}{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>, hence <math>A</math>,<math>C</math>,<math>R</math> are collineal. |

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+ | <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</math>, as requiered. | ||

==See also== | ==See also== |

## Latest revision as of 20:17, 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 LC, then =, the given information is the same as , applicating Thales theorem it follows that is parallel to .

Let be the point on such that =, in view of = and = it follows that is a parallelogram, implying that is parallel to , but we know that is parallel to , hence ,, are collineal.

is perpendicular to if and only if is the perpendicular bisector of if and only if is the angle bisector of if and only if is the angle bisector of , as requiered.