(c) \(\bigg(\frac{3+\sqrt3}{2},\frac{5-\sqrt3}{2}\bigg)\)
Let the co-ordinates of vertex C are (x, y).
Since the incentre and centroid of an equilateral triangle coincide, co-ordinates of centroid of
ΔABC = \(\bigg(\frac{9+\sqrt3}{6},\frac{15-\sqrt3}{6}\bigg)\)
∴ \(\frac{x+1+2}{3}\) = \(\frac{9+\sqrt3}{6}\) and \(\frac{2+3+y}{3}\) = \(\frac{15-\sqrt3}{6}\)
⇒ 2(x + 3) = 9 + √3 and 2(5 + y) = 15 – √3
⇒ 2x = 3 + √3 and 10 + 2y = 15 – √3
⇒ x = \(\frac{3+\sqrt3}{2}\) and 2y = 5 - √5 ⇒ y = \(\frac{5-\sqrt3}{2}\)
∴ Required co-ordinates are \(\bigg(\frac{3+\sqrt3}{2},\frac{5-\sqrt3}{2}\bigg)\).