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collinear wanted, starting with right isosceles triangle

Source: III Soros Olympiad 1996-97 R2 9.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 31, 2024
geometrycollinear

Problem Statement

Let ABCABC be an isosceles right triangle with hypotenuse ABAB, DD be some point in the plane such that 2CD=AB2CD = AB and point CC inside the triangle ABDABD. We construct two rays with a start in CC, intersecting ADAD and BDBD and perpendicular to them. On the first one, intersecting ADAD, we will plot the segment CK=ADCK = AD, and on the second one - CM=BDCM = BD. Prove that points MM, DD and KK lie on the same line.