Let ABC be a triangle with points D1, E1, F1 on the sides BC, AC, and...

   

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Let ABC be a triangle with points D1, E1, F1 on the sides BC, AC, and AB, respectively. Assume that the lines AD1, BE1,CF, intersect in a common point. Let D2 be the reflection of D1 in the bisector of the segment BC, let E2 be the reflection of E1 in the bisector of the segment AC and let F2 be the reflection of F1 in the bisector of the segment AB. Prove that the lines AD2, BE2, CF2 intersect in a common point. Let ABC be a triangle with points D1, E1, F1 on the sides BC, AC, and AB, respectively. Assume that the lines AD1, BE1,CF, intersect in a common point. Let D2 be the reflection of D1 in the bisector of the segment BC, let E2 be the reflection of E1 in the bisector of the segment AC and let F2 be the reflection of F1 in the bisector of the segment AB. Prove that the lines AD2, BE2, CF2 intersect in a common point.