1. New school for three villages. A new school is to be established to cover the area of three small villages. It is located at the point minimising the sum of squares of distances to the villages. In this question, you nd coordinates of the school given the village locations A = (a1, a2), B = (b1, b2), 0 = (01,02). (a) Let dA(:I:,y), dB(x,y), dc(a:,y) denote the distances from the point (30,31) to A, B, C, respectively. Show that the function f (x, y) := dim, y) + d2B($, y) + (1% (cc, y) has unique stationary point S. (b) Let 3A, 83, so denote the vectors from A, B, C to the point S found in part (a), respectively. Compute SA + 813 + 80. (c) For any point (my), denote by u($, y) the vector from S to (30,11). Express the vectors from A, B, C' to (30,31) in terms of SA, 83, .5C and u($, 1/). Show that aw) = ||$A||2 + IISBII2 + IISCII2 + 3||u($,y)ll2, where f (nay) is the function from part (a) and H - H is the Euclidian norm. Use this formula to explain why 8 is the global minimum of f (:c, y). [3] [2] [4] (d) Let A0 be the middle of the side BC. Compute the vector m A from A to A0 and show [2] that 3 lies on AAO. Similarly dene B0, 00 and prove that S is the intersection point of AAO, BB0, COO (medians)