Intersecting lines in a triangle

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On the sides BC, CA, AB of the triangle ABC are chosen respectively the points A1, B1, C1. A', B', C' are the middles of the segments B1C1, C1A1, A1B1 respectively. Is it true that AA', BB', CC' intersects at a common point if and only if AA1, BB1, CC1 intersects at a common point?