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# Three towns revisited

## Question

three towns are located such that they are equal distant to the center of a park. What is longer, walking from one city to the other two or walking to the park and back?

note: by taking straight paths

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I'm lazy. Let's say the radius from the park at the center to each point is 2. Draw a right triangle, center, pick a vertex, pick a side midpoint. 30:60:90 right triangle has side ratios 1:sqrt(3):2. sqrt(3) is 1/2 a side of the triangle.

Thus, 2 sides of the triangle is 4sqrt(3). City to park to city is 4. sqrt(3)>1, therefore, city-park-city is shorter.

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Couple of questions here. If the towns are all an equal distance from the park, and we're assuming this as the crow flies, where are the towns in relation to one another? If they all happen to be less than 1/3 of the diameter from each other, then the distance is going to be less than going to the park and back. But if they are also spread out more than 1/3 of the diameter of the circle from one another on average, then it would be quicker to walk to the center and back. Also, are we assuming optimal routes, if it's just strictly to each of the other towns, and not round trip?

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Couple of questions here. If the towns are all an equal distance from the park, and we're assuming this as the crow flies, where are the towns in relation to one another? If they all happen to be less than 1/3 of the diameter from each other, then the distance is going to be less than going to the park and back. But if they are also spread out more than 1/3 of the diameter of the circle from one another on average, then it would be quicker to walk to the center and back. Also, are we assuming optimal routes, if it's just strictly to each of the other towns, and not round trip?

Thank you for the question.

I meant to state that each town is equal distant to the park and equidistant to each other but the towns may not be the same distance to the park as they are to each other. The trip to the towns is one way A to B to C while going to the park is there and back.

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Aka, the perimeter of an equilateral triangle vs. twice the distance from one of the vertices to the center of the triangle.

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Aka, the perimeter of an equilateral triangle vs. twice the distance from one of the vertices to the center of the triangle.

shhhh i like to add context. lol

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Aka, the perimeter of an equilateral triangle vs. twice the distance from one of the vertices to the center of the triangle.

but not the complete perimeter. A to B to C (not back to A).

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