The circles $k_{1}$ and $k_{2}$ intersect at $A$ and $B$. The line $t$ is tangent to $k_{1}$ and $k_{2}$ at $M$ and respectively $N$. Given that $t \perp A M$ and $M N=2 A M$, calculate the measure of $\widehat{N M B}$.
Let $\mathrm{P}$ be the midpoint of [MN]. Since $\mathrm{P}$ has equal powers with respect to the circles $\mathrm{k}_{1}$ and $\mathrm{k}_{2}$, it lies on the radical axis, hence on $A B$. Since $t$ is tangent to $k_{1}, \widehat{N M B}=\widehat{M A B}$, which equals $\widehat{M A P}$ because $A, B$ and $P$ are collinear. Since triangle MAP is a right isosceles triangle, we have $\widehat{M A P}=45^{\circ}$.