The method of tangents from a point

The problem of finding the tangents to a circle $\gamma$ passing through a given point $P$ is always solved in the same way: one starts from the pencil of lines through $P$, imposes that the line of the pencil be tangent to $\gamma$, and solves with respect to $m$ (or to the parameter of the pencil).

The values found are at most two, and they correspond to:

• the two tangents from an external point;
• the single tangent from a point on the circle;
• no tangent if $P$ is internal.

Beware of the vertical line $x = x_P$: the pencil $y - y_P = m(x - x_P)$ does not contain it, so it must always be checked separately.

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Topics: Circonferenza analitica
Concepts: Fascio di rette · Punto esterno · Retta · Retta tangente
Methods: Circonferenza tangenti
Skills: Geometria analitica