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fixed point, fixed length, fixed angle (2010 Kyiv City MO Round2 10.4 11.4)

Source:

August 2, 2020
fixedgeometryFixed pointlengthanglescircumcircle

Problem Statement

The points A≠BA \ne B are given on the plane. The point CC moves along the plane in such a way that ∠ACB=α\angle ACB = \alpha , where α\alpha is the fixed angle from the interval (0o,180o0^o, 180^o). The circle inscribed in triangle ABCABC has center the point II and touches the sides AB,BC,CAAB, BC, CA at points D,E,FD, E, F accordingly. Rays AIAI and BIBI intersect the line EFEF at points MM and NN, respectively. Show that: a) the segment MNMN has a constant length, b) all circles circumscribed around triangle DMNDMN have a common point
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