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Part of 2024 Oral Moscow Geometry Olympiad

Problems(2)

KI bisects arc MN, incircle

Source: Oral Moscow Geometry Olympiad 2024, 8-9.4

9/3/2024
Given a triangle ABCABC in which the angle BB is equal to 60āˆ˜60^\circ. A circle inscribed in a triangle with a center II touches the side ACAC at point KK. A line passing through the points of touching of this circle with the other sides of the triangle intersects the its circumcircle at points MM and NN. Prove that the ray KIKI divides the arc MNMN in half.
geometry
Another construction of symedian using only ruler

Source: Oral Moscow Geometry Olympiad 2024, 10-11.4

9/3/2024
Straight lines are drawn containing the sides of an unequal triangle ABCABC, its incircle II circle and a its circumcircle, the center of which is not marked. Using only a ruler (without divisions), construct the symedian of the triangle (a straight line symmetrical to the median relative to the corresponding bisector), drawing no more than six lines.
geometry