A day at the beach

Geometry Level 5

At the Cartesian beach, the ice cream cart is at $(2,2),$ the grill is at $(8,3),$ and the beverages are sold at $(4,1).$ Where should I sit to minimize the sum of the distances to these three locations?

If the answer is $(x,y)$ , submit it as $x+y$ .

Inspiration .

The answer is 5.00.

1 solution

Michael Mendrin
Nov 28, 2015

For an acute triangle, lines drawn from point resulting minimum total distances to the vertices form angles of 120 degrees. Since the angle at (4,1) is greater than 120 degrees, that's the point .

Exactly ! For those who are not familiar with this topic, could you elaborate or provide a link, please?

Otto Bretscher - 5 years, 6 months ago

Fermat Point But an intuitive and not-so-rigorous proof is to imagine 3 pieces of string tied together at a common point, with each running through each of the vertices, and attaching equal weights to each of the other ends. Then the minimum occurs when the forces cancel out at the common point.

This is related to "soap bubble geometry", which is that when 3 planes of soap film meet at a common line, they also form angles of 120 degrees. This applies even if the films are curved.

Steiner Soap Bubbles

Here, the hiccup is that one of the string would be pulled all the way at point (4,1) There is no point outside the triangle that would meet the condition for such a Fermat Point. Imagine 3 almost colinear posts in the image of soap films---then we'd simply have two separate soap film planes meeting at a common post.

Michael Mendrin - 5 years, 6 months ago

That's a delightfully tangible explanation! Thanks!

Otto Bretscher - 5 years, 6 months ago

A day at the beach

The answer is 5.00.

1 solution

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