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Turkey NMO 2006 1st Round - P13 (Geometry)

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February 2, 2013
geometrytrigonometry

Problem Statement

Let DD be a point on the side [AB][AB] of the isosceles triangle ABCABC such that ∣AB∣=∣AC∣|AB|=|AC|. The parallel line to BCBC passing through DD intersects ACAC at EE. If m(A^)=20∘m(\widehat A) = 20^\circ, ∣DE∣=1|DE|=1, ∣BC∣=a|BC|=a, and ∣BE∣=a+1|BE|=a+1, then which of the followings is equal to ∣AB∣|AB|?
<spanclass=′latex−bold′>(A)</span> 2a<spanclass=′latex−bold′>(B)</span> a2−a<spanclass=′latex−bold′>(C)</span> a2+1<spanclass=′latex−bold′>(D)</span> (a+1)2<spanclass=′latex−bold′>(E)</span> a2+a <span class='latex-bold'>(A)</span>\ 2a \qquad<span class='latex-bold'>(B)</span>\ a^2-a \qquad<span class='latex-bold'>(C)</span>\ a^2+1 \qquad<span class='latex-bold'>(D)</span>\ (a+1)^2 \qquad<span class='latex-bold'>(E)</span>\ a^2+a
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