[The perpendicular is shorter than the oblique. Inequalities for right triangles] SLO; [Properties of the perpendicular bisectors of the sides of a triangle.

In triangle $ABC$, a point $M$ is chosen on the largest side $BC$, which is equal to $b$. Find the minimum distance between the centers of the circumcircles of triangles $BAM$ and $ACM$.
Consider the projection of the segment with endpoints at the centers of the circles onto side $B C$.

## Solution

The projections of the centers $O_{1}$ and $O_{2}$ of the given circles onto $B C$ are the midpoints $P$ and $Q$ of segments $B M$ and $M C$, respectively.

Then $O_{1} O_{2} \geqslant P Q=\frac{b}{2}$.

If $A M$ is the altitude of triangle $B A C$, then

$$
O_{1} O_{2}=P Q=\frac{b}{2}
$$

In other cases, $O_{1} O_{2}>\frac{b}{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_93426a107eaf3f76289cg-38.jpg?height=323&width=514&top_left_y=1914&top_left_x=772)

Answer

$\frac{b}{2}$.