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0232 geometry 2nd edition Round 3 p2

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May 10, 2021
geometry2nd edition

Problem Statement

Let ABCABC be a triangle with altitudes AD,BE,CFAD, BE, CF. Choose the points A1,B1,C1A_1, B_1, C_1 on the lines AD,BE,CFAD, BE, CF respectively, such that AA1AD=BB1BE=CC1CF=k.\frac{AA_1}{AD}= \frac{BB_1}{BE}= \frac{CC_1}{CF} = k. Find all values of kk such that the triangle A1B1C1A_1B_1C_1 is similar to the triangle ABCABC for all triangles ABCABC.
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