Circle tangent to two lines: the angle-bisector trick

Among the mixed conditions, tangency to two lines is handled particularly effectively with a simple geometric observation.

Observation — Circle tangent to two lines: the angle-bisector trick

If the circle is tangent to two lines $r$ and $s$, the centre $C$ lies on the bisector of the angles formed by $r$ and $s$ (because it is equidistant from $r$ and $s$, that distance being the radius). This geometric observation halves the number of unknowns and makes the problem much more manageable.

Once the bisector is identified, the centre has a single free coordinate along it: the problem, which started from three unknowns, is thus reduced to a few residual conditions.

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Topics: Circonferenza analitica
Concepts: Bisettrice · Centro · Condizioni geometriche · Distanza punto retta · Retta · Retta tangente
Skills: Geometria analitica