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2005 BMO Problems/Problem 1

Revision as of 16:19, 26 October 2018 by Dziaro44 (talk | contribs) (Created page with "Let <math>ABC</math> be an acute-angled triangle whose inscribed circle touches <math>AB</math> and <math>AC</math> at <math>D</math> and <math>E</math> respectively. Let <mat...")
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Let $ABC$ be an acute-angled triangle whose inscribed circle touches $AB$ and $AC$ at $D$ and $E$ respectively. Let $X$ and $Y$ be the points of intersection of the bisectors of the angles $C$ and $B$ with $DE$ and let $Z$ be the midpoint of $BC$. Prove that the triangle $XYZ$ is equilateral if and only if angle $A$ is equal to $60$ degrees.

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