The Cunning Midpoints

Geometry Level 4

Below lies the daunting \( \triangle ABC \).

\( A_1 \) and \( B_1 \) are the midpoints of sides \( BC \) and \( CA \) respectively.
\(AD \) is the angle bisector of \( \angle BAC \).
\( AD \) and \( A_1 B_1 \) are extended to intersect at \(K\).

Find \( A_1 K \) if \( AB = 5, BC = 12, AC = 13 \).


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