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Bulgaria 6

Source: IMO LongList 1959-1966 Problem 32

September 2, 2004
inequalitiesgeometrytriangle inequalitysimilar trianglesIMO ShortlistIMO Longlist

Problem Statement

The side lengths a,a, b,b, cc of a triangle ABCABC form an arithmetical progression (such that b−a=c−bb-a=c-b). The side lengths a1,a_{1}, b1,b_{1}, c1c_{1} of a triangle A1B1C1A_{1}B_{1}C_{1} also form an arithmetical progression (with b1−a1=c1−b1b_{1}-a_{1}=c_{1}-b_{1}). [Hereby, a=BC,a=BC, b=CA,b=CA, c=AB,c=AB, a1=B1C1,a_{1}=B_{1}C_{1}, b1=C1A1,b_{1}=C_{1}A_{1}, c1=A1B1.c_{1}=A_{1}B_{1}.] Moreover, we know that ∡CAB=∡C1A1B1.\measuredangle CAB=\measuredangle C_{1}A_{1}B_{1}.
Show that triangles ABCABC and A1B1C1A_{1}B_{1}C_{1} are similar.
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