For a △ABC , let A1 be the symmetric point of the intersection of angle bisector of ∠BAC and BC , where center of the symmetry is the midpoint of side BC, In the same way we define B1 ( on AC ) and C1 (on AB). Intersection of circumcircle of △A1B1C1 and line AB is the set {Z,C1}, with BC is the set {X,A1} and with CA is the set {Y,B1}. If the perpendicular lines from X,Y,Z on BC,CA and AB , respectively are concurrent , prove that △ABC is isosceles. geometryangle bisectorsymmetrycircumcircle