Question 26
Collinear points $A$, $B$ and $M$ are given so that $M$ is between $A$ and $B$. Let $k$ be the circle with diameter $\overline{AB}$ and let $N$ be any point on $k$ different from $A$ and $B$. Prove that the expression
$$
\frac{\operatorname{tg}(\angle ANM)}{\operatorname{tg}(\angle MAN)}
$$
is a constant, i.e. that it doesn't depend on the choice of $N$.
$$
\frac{\operatorname{tg}(\angle ANM)}{\operatorname{tg}(\angle MAN)}
$$
is a constant, i.e. that it doesn't depend on the choice of $N$.