Estonian Mathematical Olympiad

Question

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? FIGURE_0

Accepted Answer

Assume that DE is a midsegment of ABC, then D is the midpoint of AB (Fig. 24). As DP 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 FIGURE_0 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.