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concurrent wanted, 6 circles, equal into 3 pairs, tangencies given

Source: 7th QEDMO problem 7 (14. - 17. 1. 2010) https://artofproblemsolving.com/community/c1512515_qedmo_200507

May 9, 2021
geometryconcurrentconcurrencytangent circles

Problem Statement

Let ABCABC be a triangle. Let x1x_1 and x2x_2 be two congruent circles, which touch each other and the segment BCBC, and which both lie within triangle ABCABC, and for which it also holds that x1x_1 touches the segment CACA, and that x2x_2 is the segment ABAB. Let XX be the contact point of these two circles x1x_1 and x2x_2. Let y1y_1 and y2y_2 two congruent circles that touch each other and the segment CACA, and both within of triangle ABCABC, and for which it also holds that y1y_1 touches the segment ABAB, and that y2y_2 the segment BCBC. Let YY be the contact point of these two circles y1y_1 and y2y_2. Let z1z_1 and z2z_2 be two congruent circles that touch each other and the segment ABAB, and both within triangle ABCABC, and for which it also holds that z1z_1 touches the segment BCBC, and that z2z_2 the segment CACA. Let ZZ be the contact point of these two circles z1z_1 and z2z_2. Prove that the straight lines AX,BYAX, BY and CZCZ intersect at a point.
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