Incenter of a Triangle: Properties, Formulas, and Geometric Applications

incenter Chapter 6 (p. 303)

The point of concurrency of the angle bisectors of a triangle

The incenter of a triangle is the point of concurrency of the bisectors of the angles of the triangle. It is equidistant to the sides of the triangle. In other words, the center of the inscribed circle of the triangle is the incenter. The incenter is also the center of the circle that can be inscribed inside the triangle.

To find the incenter, we first draw the angle bisectors of the triangle. The incenter is the point where these angle bisectors intersect. The incenter is always inside the triangle.

The incenter is an important point of a triangle because it has several properties. Firstly, it is the center of the inscribed circle which touches all three sides of the triangle. Secondly, the radius of the inscribed circle is equal to the distance between the incenter and any one of the sides of the triangle. Thirdly, the incenter is equidistant from the three sides of the triangle.

The incenter is commonly used in solving geometric problems involving triangles. For example, the coordinates of the incenter can be used to find the equation of the circumcircle or the incircle of a triangle. Additionally, the incenter can be used to prove that the sum of the distances from the incenter to the sides of a triangle is equal to the perimeter of the triangle.

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