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Triangle inequality and symmetry

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April 23, 2009
inequalitiessymmetrygeometrygeometric transformationreflection

Problem Statement

ABCD ABCD is a 4×4 4\times 4 square. E E is the midpoint of [AB] \left[AB\right]. M M is an arbitrary point on [AC] \left[AC\right]. How many different points M M are there such that \left|EM\right|\plus{}\left|MB\right| is an integer?
<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 4<spanclass=′latex−bold′>(D)</span> 5<spanclass=′latex−bold′>(E)</span> 6<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 5 \qquad<span class='latex-bold'>(E)</span>\ 6
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