Through the midpoint D of side AB of triangle ABC, straight lines are drawn perpendicular to the bisectors

Through the midpoint D of side AB of triangle ABC, straight lines are drawn perpendicular to the bisectors of angles ABC and BAC. These lines intersect sides AC and BC at points M and K, respectively. Prove that AM = BK

Consider triangles ADM and BDK. By condition, AH and BP are the bisectors of angles A and B, as well as the perpendiculars, respectively, DM and DK, that is, they are the heights of these triangle.

If the bisector is also the height of the triangle, then such a triangle is isosceles.

Then AD = AM, BD = BK.

By hypothesis, point D is the midpoint of side AB, then AD = BD, and accordingly, then BK = AM, since the triangles are isosceles, which was required to prove.