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P09 [Geometry] - Turkish NMO 1st Round - 2002

Source:

August 8, 2014
geometryangle bisector

Problem Statement

Let ABCABC be triangle such that ∣AB∣=5|AB| = 5, ∣BC∣=9|BC| = 9 and ∣AC∣=8|AC| = 8. The angle bisector of BCA^\widehat{BCA} meets BABA at XX and the angle bisector of CAB^\widehat{CAB} meets BCBC at YY. Let ZZ be the intersection of lines XYXY and ACAC. What is ∣AZ∣|AZ|?
<spanclass=′latex−bold′>a)</span> 104<spanclass=′latex−bold′>b)</span> 145<spanclass=′latex−bold′>c)</span> 89<spanclass=′latex−bold′>d)</span> 9<spanclass=′latex−bold′>e)</span> 10 <span class='latex-bold'>a)</span>\ \sqrt{104} \qquad<span class='latex-bold'>b)</span>\ \sqrt{145} \qquad<span class='latex-bold'>c)</span>\ \sqrt{89} \qquad<span class='latex-bold'>d)</span>\ 9 \qquad<span class='latex-bold'>e)</span>\ 10
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