# Find the least expense?

You want to build a shop between three roads in the shape of an equilateral triangle.

What would be the best location for the shop so that you can reach each road with the minimum transportation cost?

Obviously, it would be inside or on the triangular roads, but I need to know how you can find the best location?

• Can you define transportation cost? Is it the sum of distances to each road?
– JS1
Sep 24, 2019 at 8:48
• Surely it depends on the SHAPE of the shop to some degree? e.g. If it can be any shape, why not make it a triangle the exact shape and size of the space between all three roads, reducing your transportation costs effectively to zero??
– Stiv
Sep 24, 2019 at 9:17
• @Stiv & JSi sorry I was not online at that time to give you reply and when I came back the question was already answered and that is the correct answer. I did not put up a lateral thinking tag. it was a logical knowledge base question. yes transportation cost means distance there you got it right JSI. Sep 24, 2019 at 14:14

## 1 Answer

Assuming "transportation cost" means sum of distances to each of the three roads, and the side of the equilateral triangle has length $$1$$:

You can build the store anywhere on or inside the triangle and the transportation cost will be $$\sqrt{3}/2$$.

Here is the proof, called Viviani's theorem

Note: I calculated the answer for a vertex, center of edge, and center of triangle and came up with the same answer each time. So then I knew it had to be a trick question. A bit of internet research led me to Viviani's theorem.

• So I did indeed misread the question. Thanks for your answer. +1 (and also +1 to the question now) Sep 24, 2019 at 11:06
• @correct if per-pendiculars are drawn to the sides of an equilateral triangle from any point in the triangle, their united length will be equal to the altitude of the triangle. Sep 24, 2019 at 14:16