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Equal product of segments

Source: China Southeast Math Olympiad 2015 Day 2 P7

June 6, 2017
geometry

Problem Statement

In △ABC\triangle ABC, we have AB>AC>BCAB>AC>BC. D,E,FD,E,F are the tangent points of the inscribed circle of △ABC\triangle ABC with the line segments AB,BC,ACAB,BC,AC respectively. The points L,M,NL,M,N are the midpoints of the line segments DE,EF,FDDE,EF,FD. The straight line NLNL intersects with ray ABAB at PP, straight line LMLM intersects ray BCBC at QQ and the straight line NMNM intersects ray ACAC at RR. Prove that PA⋅QB⋅RC=PD⋅QE⋅RFPA \cdot QB \cdot RC = PD \cdot QE \cdot RF.
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