[ Auxiliary similar triangles ] [ Properties and characteristics of isosceles triangles. ]

From the vertex $B$ of the isosceles triangle $A B C$ to its base $A C$, a height $B D$ is dropped. Each of the lateral sides $A B$ and $B C$ of triangle $A B C$ is equal to 8. In triangle $B C D$, a median $D E$ is drawn. A circle is inscribed in triangle $B D E$, touching side $B E$ at point $K$ and side $D E$ at point $M$. The segment $KM$ is equal to 2. Find the angle $A$.
As can be seen from the solution to the problem $\underline{52925}$, if the distance between the points of tangency of the inscribed circle in an isosceles triangle with the lateral sides is equal to half the lateral side, then this triangle is equilateral.

## Answer

$30^{\circ}$.