The Incircle Configuration

Tangent lengths turn sides into s − a.

The two tangents from a point to a circle are equal, so the incircle's touch points cut the sides into three tangent lengths $s-a$, $s-b$, $s-c$ — which sum to $s$ and multiply with it into Heron's formula. Translating side data into tangent-length data, and back, is the opening move of most incircle problems.

Two more handles complete the kit. The incentre angle: $\angle BIC = 90^\circ + \frac{A}{2}$, always obtuse, blind to how $B$ and $C$ split the remainder. And the arc-midpoint lemma: the midpoint $M$ of arc $BC$ satisfies $MB = MI = MC$, so $B$, $I$, $C$ lie on a circle centred at $M$ — converting incentre statements into circumcircle arcs.

The excircles obey the same equal-tangent bookkeeping with $s$ in place of $s-a$, which is why the four quantities $s$, $s-a$, $s-b$, $s-c$ keep appearing together.

More techniques

• The Extremal Principle in GraphsTake the longest path, the vertex of largest degree.
• Convexity & JensenBalance minimises what curves upward.
• All 29 techniques →