# 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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