# 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 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.