Steiner's 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 \(|A P|+|B P|+|C P|\) is a minimum. Here is how to proceed
with three points, assuming 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)
