Incircle of a triangle

A circle touching all three sides of a triangle is its incircle, and its center is the incenter¹. We find them for the triangle with vertices $A(0, 0)$ , $B(5, 0)$ and $C(3.2, 2.4)$ .

The lengths of the sides

Side	Length
$BC$	$3$
$CA$	$4$
$AB$	$5$

Since $3^2 + 4^2 = 5^2$ , the triangle is right-angled at $C$ .

How the incenter is built

The incenter is the intersection of the three internal angle bisectors. A point on the bisector of an angle is equidistant from the two sides forming it. The intersection of two bisectors is therefore equidistant from all three sides, and the remaining bisector must pass through it as well. That common distance is the radius of the incircle.

The radius from the area

Joining the incenter to the three vertices cuts the triangle into three triangles, each with one side as its base and height $r$ .

\begin{align*} S &= \frac{1}{2}ar + \frac{1}{2}br + \frac{1}{2}cr = rs \\ s &= \frac{a + b + c}{2} \end{align*}

Here $S = \dfrac{1}{2} \times 3 \times 4 = 6$ and $s = 6$ , so $r = \dfrac{S}{s} = 1$ .

For a right triangle, with $c$ the hypotenuse, the radius is also $r = \dfrac{a + b - c}{2}$ . That gives $\dfrac{3 + 4 - 5}{2} = 1$ , in agreement.

The coordinates of the incenter

The incenter is a weighted average of the vertices, weighted by the lengths of the opposite sides. Writing $a = BC$ , $b = CA$ and $c = AB$ gives the following.

I = \frac{aA + bB + cC}{a + b + c} = \frac{3(0,\, 0) + 4(5,\, 0) + 5(3.2,\, 2.4)}{12} = (3,\, 1)

The incircle is $(x - 3)^2 + (y - 1)^2 = 1$ . The $y$ -coordinate $1$ of the center is exactly its distance from the side $AB$ , so the radius can be read straight off the graph.

Difference from the circumcenter

Item	Incenter	Circumcenter
How it is found	intersection of the internal angle bisectors	intersection of the perpendicular bisectors of the sides
What it is equidistant from	the three sides	the three vertices
Position	always inside	sometimes outside

The incenter always lies inside the triangle, since being equidistant from all three sides is possible only there.

The three lines on the graph are the three sides, the two arcs are the incircle, and the large dots are the three vertices together with the incenter $(3, 1)$ .

Incircle and excircles, Wikipedia