Steiners problem for three points Given three distinct

Chapter 12, Problem 77

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Steiners problem for three points Given three distinct noncollinear points A, B, and C in the plane, find the point P in the plane such that the sum of the distances _AP_ + _BP_ + _CP_ is a minimum. Here is how to proceed with three points, assuming that the triangle formed by the three points has no angle greater than 2p>3 11202. a. Assume the coordinates of the three given points are A1x1, y12, B1x2, y22, and C1x3, y32. Let d11x, y2 be the distance between A1x1, y12 and a variable point P1x, y2. Compute the gradient of d1 and show that it is a unit vector pointing along the line between the two points. b. Define d2 and d3 in a similar way and show that _d2 and _d3 are also unit vectors in the direction of the line between the two points. c. The goal is to minimize f 1x, y2 = d1 + d2 + d3. Show that the condition fx = fy = 0 implies that _d1 + _d2 + _d3 = 0. d. Explain why part (c) implies that the optimal point P has the property that the three line segments AP, BP, and CP all intersect symmetrically in angles of 2p>3. e. What is the optimal solution if one of the angles in the triangle is greater than 2p>3 (just draw a picture)? f. Estimate the Steiner point for the three points 10, 02, 10, 12, and 12, 02.