Let ABC be a triangle. Consider the circle ωB internally tangent to the sides BC and BA, and to the circumcircle of the triangle ABC, let P be the point of contact of the two circles. Similarly, consider the circle ωC internally tangent to the sides CB and CA, and to the circumcircle of the triangle ABC, let Q be the point of contact of the two circles. Show that the incentre of the triangle ABC lies on the segment PQ if and only if AB+AC=3BC.proposed by Luis Eduardo Garcia Hernandez, Mexico geometryincentermixtilinear