If `A(0, 0), B(1, 0) and C(1/2,sqrt(3)/2)` then the centre of the circle for which the lines `AB,BC, CA` are tangents is

Options

A. `((1)/(2),(1)/(4))`
B. `((3)/(2),(sqrt3)/(2))`
C. `((1)/(2),(1)/(2sqrt3))`
D. `((1)/(2),-(1)/(sqrt3))`

Answer

C

Explanation

`AB=AC=BC=1`
So, triangle is an equilateral.
Thus, incentre coincide with centroid.
Centre of the circle touching all the three sides of triangle is incentre.
therefore, incentre is
`((0+1+(1)/(2))/3,(0+0+(sqrt3)/(2))/3)=((1)/(2),(1)/(2sqrt3))`