Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.6 AIME, harder Prove it Estonia

Points $P$ and $Q$ are chosen on the side $BC$ of triangle $ABC$ in such a way that $P$ lies between $B$ and $Q$ , and rays $AP$ and $AQ$ trisect the angle $BAC$ . The line parallel to $AQ$ and passing through $P$ meets the side $AB$ of the triangle at point $D$ , and the line parallel to $AP$ and passing through $Q$ meets the side $AC$ of the triangle at point $E$ . Can it happen that $DE$ is a midsegment of the triangle $ABC$ ?

Solution

Assume that $DE$ is a midsegment of $ABC$ , then $D$ is the midpoint of $AB$ (Fig. 24). As $DP \parallel AQ$ , $DP$ is a midsegment of triangle $ABQ$ . Hence, $P$ is a midpoint of $BQ$ and $AP$ is a median of triangle $ABQ$ . As rays $AP$ and $AQ$ trisect the angle $BAC$ , $AP$ is a bisector of angle $QAB$ . Thus, $ABQ$ is an isosceles triangle with altitude $AP$ , implying that $AP$ is perpendicular to $BC$ . Similarly we can

Fig. 24

see that $AQ$ is perpendicular to $BC$ . This leads to a contradiction as $AP$ and $AQ$ cannot coincide. Therefore, $DE$ cannot be a midsegment of $ABC$ .

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