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This essential trains for: AMC-10, AMC-12, AIME, Math Kangaroo 9-10.

The incircle of a triangle is the circle that is tangent to all the sides of the triangle. Since three non-collinear points in the plane determine a circle uniquely, the points of tangency determine only one incircle for any triangle. The center of the incircle (incenter) is found at the intersection of the angle bisectors, since it has to be equidistant from the sides of the triangle. The radius of the incircle (inradius) can be calculated as a function of the area and perimeter of the triangle.

Let the circle with center O be the incircle of scalene triangle ABC and let the side lengths be denoted with a, b, c as in the figure.

The radius of the circumcircle is traditionally denoted with r.

Due to the fact that the radius of a circle is perpendicular to a tangent line at the point of tangency, we can dissect the triangle ABC in three triangles that have altitudes equal to r: AOB, BOC, and COA:

The area of triangle ABC can be written as the sum of the areas of the three triangles that form the dissection:

Factoring and cleaning up we obtain:

Denote the half of the sum of the side lengths as the semiperimeter:

and solve for the inradius: