GeoProof — Incenter & Incircle explorer

The incenter is equidistant from all three sides — that distance is the inradius r

The Incenter & Incircle

An angle bisector cuts a corner of the triangle into two equal angles. A triangle has three of them, and they all pass through one point — the incenter I:

three angle bisectors meet at the incenter I

The incenter is special because it is the same perpendicular distance from all three sides. That distance is the inradius r, and the circle centered at I with radius r — the incircle — touches (is tangent to) all three sides from the inside.

Why equidistant?

Any point on the bisector of an angle is equally far from the two sides forming that angle. A point on all three bisectors is therefore equally far from all three sides — so such a point exists and is unique.

A handy formula

The inradius relates the triangle's area and semiperimeter by r = Area ÷ s, where s = (a + b + c) / 2. So the area also equals r · s — the incircle "spreads" the area evenly along the perimeter.

Always inside

Unlike the circumcenter or orthocenter, the incenter is always inside the triangle, even for obtuse triangles — drag a vertex to make a very flat triangle and watch I stay within.

Where it's used

The incircle is the largest circle that fits inside the triangle. Incenters appear throughout geometry — in tangential polygons (see Pitot's theorem), constructions, and optimization problems.