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2019 Serbia MO Day 2 P4

Source: 2019 Serbia MO

April 7, 2019
geometryangle bisectorsymmetrycircumcircle

Problem Statement

For a △ABC\triangle ABC , let A1A_1 be the symmetric point of the intersection of angle bisector of ∠BAC\angle BAC and BCBC , where center of the symmetry is the midpoint of side BCBC, In the same way we define B1B_1 ( on ACAC ) and C1C_1 (on ABAB). Intersection of circumcircle of △A1B1C1\triangle A_1B_1C_1 and line ABAB is the set {Z,C1}\{Z,C_1 \}, with BCBC is the set {X,A1}\{X,A_1\} and with CACA is the set {Y,B1}\{Y,B_1\}. If the perpendicular lines from X,Y,ZX,Y,Z on BC,CABC,CA and AB AB , respectively are concurrent , prove that △ABC\triangle ABC is isosceles.
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