the inscribed circle

Now flip the previous question inside out: instead of a circle wrapping around a triangle through its corners, find the largest circle that fits snugly INSIDE the triangle, touching all three sides without crossing any. That is the inscribed circle, and again there is exactly one. The task is to find its centre and radius by construction.

The centre, called the incentre, is the one point equidistant from all three sides (not the vertices). The angle bisector of an angle is exactly the set of points equidistant from its two sides. So bisect angle A and angle B; their intersection I is equidistant from all three sides at once. For the radius, drop a perpendicular from I to any side and measure to the foot — that distance r is the inradius, the same to every side. Draw the circle of radius r centred at I and it touches all three sides as a tangent.

Where the circumcentre came from perpendicular bisectors and may wander outside the triangle, the incentre comes from angle bisectors and always sits safely inside. The point where the incircle touches a side is the foot of the perpendicular from the incentre, and the radius there is perpendicular to the side (tangent-radius perpendicularity). Inscribed circles appear in design, packing, and any setting where you want the roundest object a triangle can hold.