All triangles have an incenter, and it always lies inside the triangle. One way to find the incenter makes use of the property that the incenter is the intersection of the three angle bisectors, using coordinate geometry to determine the incenter's location. Unfortunately, this is often computationally tedious.
Generally, the easiest way to find the incenter is by first determining the inradius, or radius of the incircle, usually denoted by the letter (the letter is reserved for the circumradius). This can be done in a number of ways, detailed in the 'Basic properties' section below. Once the inradius is known, each side of the triangle can be translated by the length of the inradius, and the intersection of the resulting three lines will be the incenter. This, again, can be done using coordinate geometry.
Alternatively, the following formula can be used. For a triangle with side lengths , with vertices at the points , the incenter lies at
A triangle has vertices at , and . What are the coordinates of the incenter?
The lengths of the sides (using the distance formula) are Now the above formula can be used:
The simplest proof is a consequence of the trigonometric version of Ceva's theorem, which states that concur if and only if
In this case, are the feet of the angle bisectors, so , , and . As a result,
and rearranging the left hand side gives
Therefore, the three angle bisectors intersect at a single point, .
Furthermore, since lies on the angle bisector of , the distance from to is equal to the distance from to . Similarly, this is also equal to the distance from to . Therefore, is the center of the inscribed circle, proving the existence of the incenter.
It's been noted above that the incenter is the intersection of the three angle bisectors.
For a triangle with semiperimeter (half the perimeter) and inradius ,
The area of the triangle is equal to .
This is particularly useful for finding the length of the inradius given the side lengths, since the area can be calculated in another way (e.g. Heron's formula), and the semiperimeter is easily calculable.
As a corollary,
In a right triangle with integer side lengths, the inradius is always an integer.
If is the point where the incircle touches , and similarly are where the incircle touches and respectively, then . As a corollary,
, and also
Furthermore and intersect at a single point, called the Gergonne point.
If the altitudes of a triangle have lengths , then
Let the sides of the triangle be . Then, since , and similarly for other sides,
The incircle and circumcircle are also intimately related. According to Euler's theorem,
where is the circumradius, the inradius, and the distance between the incenter and the circumcenter. Equivalently, . This also proves Euler's inequality: . Equality holds only for equilateral triangles.
Incircles also relate well with themselves. If are the radii of the three circles tangent to the incircle and two sides of the triangle, then
On a different note, if the circumcircle of is drawn, and is the midpoint of minor arc , then
is also the circumcenter of .
Equivalently, . This is known as "Fact 5" in the Olympiad community.
Similarly, if point lies on the circumcircle of so that , then . is also perpendicular to , where is the circumcenter of .
All triangles have an incircle, and thus an incenter, but not all other polygons do. When one exists, the polygon is called tangential. As in a triangle, the incenter (if it exists) is the intersection of the polygon's angle bisectors.
In the case of quadrilaterals, an incircle exists if and only if the sum of the lengths of opposite sides are equal: