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midpoint wanted, CD = CK, incircle related

Source: 2021 Saudi Arabia Training Lists p18 https://artofproblemsolving.com/community/c2758131_2021_saudi_arabia_training_tests

January 6, 2022
geometrymidpointincircle

Problem Statement

Let ABCABC be a triangle with AB<ACAB < AC and incircle (I)(I) tangent to BCBC at DD. Take KK on ADAD such that CD=CKCD = CK. Suppose that ADAD cuts (I)(I) at GG and BGBG cuts CKCK at LL. Prove that K is the midpoint of CLCL.
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