Construction: Tangents from a Point

Suppose that we have a circle $\Omega$ and a point $P.$ We wish to construct the lines that pass through $P$ and are tangent to $\Omega.$

The first step is to construct the center $O$ of $\Omega.$ This is easy enough; one way to do this is to pick any $3$ points $A, B, C$ on $\Omega$ and then construct the perpendicular bisectors of any $2$ of the $3$ sides of $\triangle ABC.$

The second step is to construct the midpoint $M$ of $OP.$ This can be done by constructing the perpendicular bisector of segment $OP.$

The third step is to draw the circle $\omega$ with center $M$ and radii $MO$ and $MP.$ Let $X$ and $Y$ be the intersections of $\Omega$ and $\omega.$

The fourth and final step is to draw the lines $PX$ and $PY.$ These are the tangent lines to $\Omega$ passing through $P.$

Why does this construction work? The explanation is simple enough. Since $M$ is the midpoint of segment $OP,$ segment $OP$ is a diameter of $\omega.$ Thus, $\angle OXP$ and $\angle OYP$ are both right angles. It follows that $PX$ and $PY$ are tangent to $\Omega.$