Let $z$ be a complex number such that
\[|z - 12| + |z - 5i| = 13.\]Find the smallest possible value of $|z|.$
I notice that the equation involves the sum of two distances: the distance from $z$ to $12$ and the distance from $z$ to $5i$. This makes me think of the triangle inequality, which says that the sum of the lengths of any two sides of a triangle is greater than or equal to the length of the third side. So I can imagine that $z$, $12$, and $5i$ are the vertices of a triangle, and the equation says that the sum of two sides is exactly $13$. That means the triangle must be degenerate, or a straight line, since the equality holds. So $z$ must lie on the line segment that joins $12$ and $5i$. Now I want to find the point on that line segment that is closest to the origin, since that would minimize $|z|$. I can use the formula for the distance from a point to a line to find that point. The equation of the line passing through $12$ and $5i$ is $y = \frac{5}{12}x - 5$, so the distance from $(0,0)$ to this line is $\frac{|0 - 0 - 5|}{\sqrt{1 + (\frac{5}{12})^2}} = \frac{60}{13}$. This means that the point on the line that is closest to the origin has a distance of $\frac{60}{13}$ from the origin, and this is the smallest possible value of $|z|$. 