[ Properties and characteristics of an isosceles triangle. ]

The median $A M$ of triangle $A B C$ is perpendicular to its bisector $B K$. Find $A B$, if $B C=12$.
Let $P$ be the intersection point of $B K$ and $A M$. In triangle $A B M$, the bisector $B P$ is an altitude.

## Solution

Let $P$ be the intersection point of segments $B K$ and $A M$. In triangle $A B M$, the bisector $B P$ is an altitude, so triangle $A B M$ is isosceles. Therefore, $A B = B M = 1 / 2 B C = 6$.

## Answer

6.