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tangent wanted, sstrarin with circumcircle

Source: 2005 All-Ukrainian Correspondence MO of magazine ''In the World of Mathematics'', grades 5-11 p11

April 28, 2021
geometrytangentcircumcircleUkraine Correspondence

Problem Statement

Let the circle ω\omega be circumscribed around the triangle △ABC\vartriangle ABC with right angle ∠A\angle A. Tangent to the circle ω\omega at point AA intersects the line BCBC at point DD. Point EE is symmetric to AA with respect to the line BCBC. Let KK be the foot of the perpendicular drawn from point AA on BEBE, LL the midpoint of AKAK. The line BLBL intersects the circle ω\omega for the second time at the point NN. Prove that the line BDBD is tangent to the circle circumscribed around the triangle △ADM\vartriangle ADM.
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