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Prove that angle DBQ is right

Source: Sharygin First Round 2013, Problem 7

April 8, 2013
geometryincenterAsymptoteprojective geometrygeometry proposed

Problem Statement

Let BDBD be a bisector of triangle ABCABC. Points IaI_a, IcI_c are the incenters of triangles ABDABD, CBDCBD respectively. The line IaIcI_aI_c meets ACAC in point QQ. Prove that ∠DBQ=90∘\angle DBQ = 90^\circ.
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