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Midpoint of the segment joining two excentres of a triangle

Source: MEMO 2017 T5

August 25, 2017
geometry

Problem Statement

Let ABCABC be an acute-angled triangle with AB>ACAB > AC and circumcircle Γ\Gamma. Let MM be the midpoint of the shorter arc BCBC of Γ\Gamma, and let DD be the intersection of the rays ACAC and BMBM. Let E≠CE \neq C be the intersection of the internal bisector of the angle ACBACB and the circumcircle of the triangle BDCBDC. Let us assume that EE is inside the triangle ABCABC and there is an intersection NN of the line DEDE and the circle Γ\Gamma such that EE is the midpoint of the segment DNDN. Show that NN is the midpoint of the segment IBICI_B I_C, where IBI_B and ICI_C are the excentres of ABCABC opposite to BB and CC, respectively.
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