Tangent at a point of the circle

When the point from which the tangent is drawn lies on the circle, the problem simplifies: there is a single tangent and it can be found geometrically, exploiting the property that the tangent is perpendicular to the radius at the point of contact.

Observation — Tangent at a point of the circle

If the point $T$ lies on the circle, there is a single tangent and it can also be found in a “geometric” way: the tangent is perpendicular to the radius $CT$ at the point $T$. One computes the gradient of $CT$, takes its negative reciprocal and writes the line through $T$.

This avoids the system with the circle altogether: the gradient of $CT$ and the perpendicularity condition $m_{\text{tan}} = -\dfrac{1}{m_{CT}}$ are enough.

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Topics: Circonferenza analitica
Concepts: Coefficiente angolare · Perpendicolarita · Raggio · Retta · Retta tangente
Methods: Circonferenza tangenti
Skills: Geometria analitica