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collinear wanted, starting with incircle of isosceles triangle

Source: 2016 All-Ukrainian Correspondence MO of magazine ''In the World of Mathematics'', grades 5-12 p7

April 27, 2021
geometrycollinearincircleisoscelesUkraine Correspondence

Problem Statement

The circle ω\omega inscribed in an isosceles triangle ABCABC (AC=BCAC = BC) touches the side BCBC at point DD .On the extensions of the segment ABAB beyond points AA and BB, respectively mark the points KK and LL so that AK=BLAK = BL, The lines KDKD and LDLD intersect the circle ω\omega for second time at points GG and HH, respectively. Prove that point AA belongs to the line GHGH.