MathDB

4

Part of 1990 Turkey Team Selection Test

Problems(1)

That area problem

Source: Turkey TST 1990 - P4

9/11/2013
Let ABCDABCD be a convex quadrilateral such that E,F∈[AB],AE=EF=FBG,H∈[BC],BG=GH=HCK,L∈[CD],CK=KL=LDM,N∈[DA],DM=MN=NA\begin{array}{rl} E,F \in [AB],& AE = EF = FB \\ G,H \in [BC],& BG = GH = HC \\ K,L \in [CD],& CK = KL = LD \\ M,N \in [DA],& DM = MN = NA \end{array} Let [NG]∩[LE]={P},[NG]∩[KF]={Q},[NG] \cap [LE] = \{P\}, [NG]\cap [KF] = \{Q\}, [MH]∩[KF]={R},[MH]∩[LE]={S}{[}MH] \cap [KF] = \{R\}, [MH]\cap [LE]=\{S\} Prove that [*]Area(ABCD)=9⋅Area(PQRS)Area(ABCD) = 9 \cdot Area(PQRS) [*] NP=PQ=QGNP=PQ=QG
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