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Angle Bisector Theorem problem involving triangle ABC with integer side lengths and a given ratio.

Diagram illustrating the Angle Bisector Theorem with triangle ABC, where BD bisects angle ABC, AD = 3, DC = 8, and side lengths AB = 6, BC = 16, and AC = 11, demonstrating the theorem's application to find the smallest possible perimeter.

Diagram illustrating the Angle Bisector Theorem with triangle ABC, where BD bisects angle ABC, AD = 3, DC = 8, and side lengths AB = 6, BC = 16, and AC = 11, demonstrating the theorem's application to find the smallest possible perimeter.

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