What are the properties of the incenter of a triangle?

The incenter is the center of the largest circle that can be inscribed inside the circle.

Answer: Incenter is the center of the biggest circle that can be inscribed within the circle.

An incenter can never lie outside the triangle.

Explanation:

An incenter of a triangle is the point where three angle bisectors of a triangle meet. Also, referred to as one of the points of triangle concurrency.

The incenter is the center of the triangle's incenter - the largest circle that will fit inside the triangle.

Look at the properties of the incenter.

1. Center of the incircle.
2. Always lies inside the triangle.
3. If you link the incenter to two edges perpendicularly, and the included vertex you will see a pair of congruent triangles.
4. It is equidistant from the sides of the triangle.

Here is the formula to find the incircle radius.

Let the radius of the incircle be r and all the sides of the triangle be a, b, c.

r = √ [(s-a)(s-b)(s-c)/s]

Here, s = semiperimeter of the triangle = (a + b + c) / 2

Coordinates of the incenter of the triangle:

It is easy to locate the vertices of the incenter if we know the coordinates of the vertices of the triangle and its lengths.

Let us suppose the coordinates of the vertex of the triangle to be (X_A, Y_A),  (X_B, Y_B) and (X_C, Y_C)

Therefore, coordinates of the incenter are:

x1 = (x_A × a + x_B × b + x_C × c) / (a + b + c)

y1 = (y_A × a + y_B × b + y_C × c) / (a + b + c)