Problem 9.6. Given a triangle $ABC$, point $M$ is the midpoint of side $BC$. Let $\ell$ be the bisector of the external angle $A$ of triangle $ABC$. The line passing through $M$ and parallel to $\ell$ intersects side $AB$ at point $K$. Find the length of segment $AK$ if $AB=23$ and $AC=8$.

![](https://cdn.mathpix.com/cropped/2024_05_06_f585fb32d0f8951822d4g-6.jpg?height=429&width=521&top_left_y=1144&top_left_x=466)
Answer: 15.5.

Solution. Draw a line $\ell_{1}$ through point $C$ parallel to line $\ell$ (Fig. 4). Let $X$ be the point of intersection of line $\ell_{1}$ and $AB$.

Notice that $\angle AXC = \angle ACX$, since both these angles are equal to half of the external angle $A$, so $AX = AC = 8$.

![](https://cdn.mathpix.com/cropped/2024_05_06_f585fb32d0f8951822d4g-7.jpg?height=430&width=517&top_left_y=84&top_left_x=468)

Fig. 4: Solution to problem 9.6

In triangle $BCX$, segment $MK$ is the midline, so

$$
AK = AX + XK = 8 + \frac{BX}{2} = 8 + \frac{AB - AX}{2} = 8 + \frac{23 - 8}{2} = 15.5
$$