Vector AB=M'C-MA. This vector is perpendicular to the MA (tangent). I don 't understand why? I get why the angles MOM' and AMB are equal because they have their sides perpendicular, but why MAB is a right angle?
MAB is an isosceles triangle because its two sides MA and MB are the same length (unit vectors). Since it is an isosceles triangle, the angle MAB = the angle MBA. The sum of the angles in the triangle is $d\theta + 2 (MAB)=180\deg$. As $d \theta \rightarrow 0$ then $MAB \rightarrow 90\deg$.