27. In triangle $A B C$, a perpendicular line passing through the midpoint of side $A B$ intersects line $A C$ at point $M$, and a perpendicular line passing through the midpoint of $A C$ intersects line $A B$ at point $N$. It is known that $|M N|=|B C|$ and line $M N$ is perpendicular to line $B C$. Determine the angles of triangle $A B C$.
27. Let $|BC|=a, \hat{C}>\hat{B}, D$ and $E$ be the midpoints of $AB$ and $AC$. The quadrilateral $EMDN$ is cyclic (since $\widehat{MEN}=\widehat{MDN}=\left.=90^{\circ}\right), |MN|=a, |ED|=\frac{a}{2}, MN$ is the diameter of the circle circumscribed around $MEND$. Therefore, $\widehat{DME}=30^{\circ}, \widehat{CAB}=$ $=90^{\circ}-\widehat{EMD}=60^{\circ}, \quad \widehat{CBA}=\widehat{EDN}=\widehat{EMN}=\frac{1}{2} \widehat{EMD}=15^{\circ}$, $\widehat{ACB}=105^{\circ}$.

Answer: $\hat{A}=60^{\circ}, \hat{B}=15^{\circ}, \hat{C}=105^{\circ}$ or $\hat{A}=60^{\circ}, \hat{B}=105^{\circ}$, $\hat{C}=15^{\circ}$.