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S_PQR/S_A'B'C' - Iran NMO 1999 (Second Round) Problem5

Source:

October 4, 2010
geometry proposedgeometry

Problem Statement

Let ABCABC be a triangle and points P,Q,RP,Q,R be on the sides AB,BC,ACAB,BC,AC, respectively. Now, let A′,B′,C′A',B',C' be on the segments PR,QP,RQPR,QP,RQ in a way that AB∣∣A′B′AB||A'B' , BC∣∣B′C′BC||B'C' and AC∣∣A′C′AC||A'C'. Prove that: ABA′B′=SPQRSA′B′C′. \frac{AB}{A'B'}=\frac{S_{PQR}}{S_{A'B'C'}}. Where SXYZS_{XYZ} is the surface of the triangle XYZXYZ.
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