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Extend DP until it cuts the circumcircle again at N

Source: China TST 1993, problem 6

June 27, 2005
geometrycircumcircleincentertrigonometrygeometric transformationreflectionRing Theory

Problem Statement

Let ABCABC be a triangle and its bisector at AA cuts its circumcircle at D.D. Let II be the incenter of triangle ABC,ABC, MM be the midpoint of BC,BC, PP is the symmetric to II with respect to MM (Assuming PP is in the circumcircle). Extend DPDP until it cuts the circumcircle again at N.N. Prove that among segments AN,BN,CNAN, BN, CN, there is a segment that is the sum of the other two.
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