Question 172
Let $\Gamma$ be a circle with centre $O$ and radius $r$, and let $\ell$ be a line that does not meet $\Gamma$. Let $E$ be the intersection point of $\ell$ and the line perpendicular to $\ell$ through $O$.
Let $M$ be a point of $\ell$ different from $E$. The tangents to the circle $\Gamma$ through $M$ touch $\Gamma$ at $A$ and $B$. Finally, let $H$ be the intersection point of the lines $AB$ and $OE$.
Prove that $OH = \dfrac{r^2}{OE}$.
Let $M$ be a point of $\ell$ different from $E$. The tangents to the circle $\Gamma$ through $M$ touch $\Gamma$ at $A$ and $B$. Finally, let $H$ be the intersection point of the lines $AB$ and $OE$.
Prove that $OH = \dfrac{r^2}{OE}$.