The incentre of a triangle with vertices (0, 0), (1, √3) and (2, 0) is: 

Concept:

In-center: The in-centre of a triangle is a point inside the triangle which is equidistant from all the three sides. Equivalently, it is the point of intersection of the three angle bisectors of the triangle.

Co-ordinates of the in-center: If (x₁, y₁), (x₂, y₂) and (x₃, y₃) be the three vertices of a triangle and a, b, c be the lengths of the sides opposite to these vertices respectively, then the co-ordinates of the in-center are given by: \(\rm \left ( \dfrac{ax_1+bx_2+cx_3}{a+b+c}, \dfrac{ay_1+by_2+cy_3}{a+b+c} \right)\).

Calculation:

Let (x1, y1) = (0, 0), (x2, y2) = (1, √3) and (x3, y3) = (2, 0).

∴ a = √[(1 - 2)2 + (√3 - 0)2] = √(1 + 3) = 2

b = √(4 + 0) = 2

c = √(1 + 3) = 2

Since the given triangle is an equilateral triangle(a = b = c). Therefore, the incentre coincides with the centroid.

\(\rm \left ( \dfrac{x_1+x_2+x_3}{3}, \dfrac{y_1+y_2+y_3}{3} \right )= \left ( \dfrac{0+1+2}{3}, \dfrac{0+\sqrt 3+0}{3} \right )= \left ( \dfrac{1}{3}, \dfrac{1}{\sqrt3} \right )\).