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ВС/US = DE/SТ = FА/ТU if АВ/РR = CD/RQ= ЕF/QP, convex hexagon ABCDEF

Source: 2003 VIII All-Ukrainian Correspondence MO of magazine ''In the World of Mathematics'', grades 5-11 p11

January 10, 2023

geometryhexagonratio

Problem Statement

Let

ABCDEF

be a convex hexagon,

P, Q, R

be the intersection points of

AB

and

EF

EF

and

CD

CD

and

AB

S, T,UV

are the intersection points of

BC

and

DE

DE

and

FA

FA

and

BC

, respectively. Prove that if

\frac{AB}{PR}=\frac{CD}{RQ}=\frac{EF}{QP},

then

\frac{BC}{US}=\frac{DE}{ST}=\frac{FA}{TU}.