8,9
In triangle $ABC$, the median $AM$ is extended beyond point $M$ by a distance equal to $AM$.
Find the distance from the resulting point to vertices $B$ and $C$, if $AB=4$, $AC=5$.
Let $A_{1}$ be a point on the extension of the median $A M$ beyond point $M$, such that $M A_{1}=A M$. Prove the equality of triangles $A_{1} M B$ and $A M C$.

## Solution

Let $A_{1}$ be a point on the extension of the median $A M$ beyond point $M$, such that $M A_{1}=A M$. Triangles $A_{1} M B$ and $A M C$ are equal by two sides and the angle between them, so $A_{1} B=A C=5$. Similarly, $A_{1} C=A B=4$.
5 and 4.
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