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similar triangles

Source: Vietnam NMO 1995, Problem 3

September 7, 2008
geometry proposedgeometry

Problem Statement

Let a non-equilateral triangle ABC ABC and AD,BE,CF AD,BE,CF are its altitudes. On the rays AD,BE,CF, AD,BE,CF, respectively, let A′,B′,C′ A',B',C' such that \frac {AA'}{AD} \equal{} \frac {BB'}{BE} \equal{} \frac {CC'}{CF} \equal{} k. Find all values of k k such that △A′B′C′∼△ABC \triangle A'B'C'\sim\triangle ABC for any non-triangle ABC. ABC.
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