2. Let ABC be a triangle. Construct equilateral triangles ABC", ACB', and BCA outside ABC as in Figure 1. (a) Show that the gravity centres of the triangles ABC and A'B'C' coincide. (b) Let U, V, and W be the gravity centres of the triangles BCA', ACB, respec- tively ABC'. Show that the triangle UVW is equilateral. Hint. Denote by a, b, c, d',V, d,u,v, and w the complex numbers corresponding to the points A, B, C, A', B', C',U,V, respectively W. You'll have to find a',V, d,u,v, and w in terms of a, b, and c. To this end, note that C' is obtained from A by a rotation about B through /3. So db = E. a-b Here e = cos + i sin 5. Get d' from the previous equation. Similarly you will get b' and a'. For point (a), use question 1 above. For point (b), verify that u-v= = E(W - U) (you'll need that ? - +1 = 0). B'b') A) C'le'). VW Ww) B(b) B) Cle) Uu) Ale") Figure 1 2. Let ABC be a triangle. Construct equilateral triangles ABC", ACB', and BCA outside ABC as in Figure 1. (a) Show that the gravity centres of the triangles ABC and A'B'C' coincide. (b) Let U, V, and W be the gravity centres of the triangles BCA', ACB, respec- tively ABC'. Show that the triangle UVW is equilateral. Hint. Denote by a, b, c, d',V, d,u,v, and w the complex numbers corresponding to the points A, B, C, A', B', C',U,V, respectively W. You'll have to find a',V, d,u,v, and w in terms of a, b, and c. To this end, note that C' is obtained from A by a rotation about B through /3. So db = E. a-b Here e = cos + i sin 5. Get d' from the previous equation. Similarly you will get b' and a'. For point (a), use question 1 above. For point (b), verify that u-v= = E(W - U) (you'll need that ? - +1 = 0). B'b') A) C'le'). VW Ww) B(b) B) Cle) Uu) Ale") Figure 1