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Part of 2014 Turkey MO (2nd round)

Problems(1)

Prove that the two lines are perpendicular

Source: Turkey National Olympiad 2014 P3

11/17/2014
Let D,E,FD, E, F be points on the sides BC,CA,ABBC, CA, AB of a triangle ABCABC, respectively such that the lines AD,BE,CFAD, BE, CF are concurrent at the point PP. Let a line ℓ\ell through AA intersect the rays [DE[DE and [DF[DF at the points QQ and RR, respectively. Let MM and NN be points on the rays [DB[DB and [DC[DC, respectively such that the equation QN2DN+RM2DM=(DQ+DR)2−2⋅RQ2+2⋅DM⋅DNMN \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} holds. Show that the lines ADAD and BCBC are perpendicular to each other.
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