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| Medially Similar (Posted on 2005-08-19) |
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Let ABC be a triangle, with M the midpoint of BC, and DEF a triangle, with N the midpoint of EF. Suppose that angle BAM equals angle EDN and angle CAM equals angle FDN. Show that triangles ABC and DEF are similar.
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Submitted by McWorter
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Solution:
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Without loss we may assume that |AM|=|DN| and superimpose triangle DEF onto triangle ABC so that M and N coincide, E is on AB, and F is on AC. Since |BM|=|MC| and |EN|=|NF|, triangles BEM (equal BEN) and CFM (equal CFN) are congruent. Hence angles EBM and MCF are equal, whence BE and CF are parallel. This is a contradiction unless B and E coincide, and so also C and F coincide. Hence triangles ABC and DEF are similar. |
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