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Turkish NMO First Round - 2012 Problem - 21 {Geometry}

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July 1, 2012
geometryangle bisector

Problem Statement

The angle bisector of vertex AA of △ABC\triangle ABC cuts [BC][BC] at DD. The circle passing through AA and touching to BCBC at DD meets [AB][AB] and [AC][AC] at PP and QQ, respectively. ADAD and PQPQ meet at TT. If ∣AB∣=5,∣BC∣=6,∣CA∣=7|AB|=5, |BC|=6, |CA|=7, then ∣AT∣∣TD∣=?\frac{|AT|}{|TD|}=?
<spanclass=′latex−bold′>(A)</span> 75<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 72<spanclass=′latex−bold′>(E)</span> 4 <span class='latex-bold'>(A)</span>\ \frac75 \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ 3 \qquad <span class='latex-bold'>(D)</span>\ \frac72 \qquad <span class='latex-bold'>(E)</span>\ 4
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