Incenter of a Triangle: Definition, Formula, and Applications in Geometry

Incenter

The point of intersection of the angle bisectors of a triangle is called the incenter.

Incenter is the point of concurrency of the three angle bisectors of a triangle. It is the center of the circle inscribed in the triangle, which is the circle that is tangent to all three sides of the triangle.

To find the incenter of a triangle, we need to follow these steps:

1. Draw the triangle and mark the intersections of the angle bisectors as points A, B, and C.

2. Draw the lines from A, B, and C that are perpendicular to the opposite sides of the triangle. These lines will intersect at the incenter.

3. The incenter is equidistant from all three sides of the triangle. This means that the distance from the incenter to any side of the triangle is equal to the radius of the inscribed circle.

4. To find the coordinates of the incenter, we can use the formula:

Incenter = (ax + bx + cx)/ (a + b + c), (ay + by + cy)/ (a + b + c)

where (ax, ay), (bx, by), and (cx, cy) are the coordinates of the vertices of the triangle, and a, b, and c are the lengths of the sides opposite those vertices.

Incenter is an important geometric point as it is the center of the circle inscribed in the triangle, which has many applications in geometry. For example, it helps to find the area of the triangle using the formula:

Area = s * r

where s is the semiperimeter of the triangle and r is the inradius (radius of the inscribed circle).

More Answers: