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 \(|A P|+|B P|+|C
P|\) 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 \(2 \pi /
3\left(120^{\circ}\right)\).
a. Assume the coordinates of the three given points are \(A\left(x_{1},
y_{1}\right)\) \(B\left(x_{2}, y_{2}\right),\) and \(C\left(x_{3}, y_{3}\right) .\)
Let \(d_{1}(x, y)\) be the distance between \(A\left(x_{1}, y_{1}\right)\) and a
variable point \(P(x, y) .\) Compute the gradient of \(d_{1}\) and show that it is
a unit vector pointing along the line between the two points.
b. Define \(d_{2}\) and \(d_{3}\) in a similar way and show that \(\nabla d_{2}\)
and \(\nabla d_{3}\) are also unit vectors in the direction of the line between
the two points.
c. The goal is to minimize \(f(x, y)=d_{1}+d_{2}+d_{3}\) Show that the condition
\(f_{x}=f_{y}=0\) implies that \(\nabla d_{1}+\nabla d_{2}+\nabla d_{3}=0\).
d. Explain why part (c) implies that the optimal point \(P\) has the property
that the three line segments \(A P, B P,\) and \(C P\) all intersect symmetrically
in angles of \(2 \pi / 3\).
e. What is the optimal solution if one of the angles in the triangle is
greater than \(2 \pi / 3\) (just draw a picture)?
f. Estimate the Steiner point for the three points (0,0),(0,1) and (2,0)
