The equation of the circumcircle of an equilateral triangle is x2+y2+2gx+2fy+c=0 and one vertex of thetriangle is (1, 1). The equation of the incircle of the triangle, is

In an equilateral triangle the circumcentre and incentre coincide. So, the coordinates of the incentre are (- g, - f). Also, in an equilateral triangle, circumradius is twice the in-radius.∴ In- radius =12 (Circum- radius) =12 82+f2−cBut, the circumcircle passes through (1, 1). Therefore1+1+2g+2f+c=0⇒c=−2−2g−2f∴ In-radius =12g2+f2+2+2g+2f=12(g+1)2+(f+1)2Hence, the equation of the in-circle is(x+g)2+(y+f)2=14(g+1)2+(f+1)2⇒ 4x2+y2+8gx+8fy=1+2g−3g2+1+2f−3f2⇒ 4x2+y2+8gx+8fy=(1−g)(1+3g)+(1−f)(1+3f)