Pigeonhole Principle?

[Littlelad]

Im pretty sure the answer is no but i only got this by drawing an equaltioral triangle Think i am supposed to use the pigeonholre principle Any ideas?????

Five Professors want to built their houses within the triangular area between the Uni-
versity, the shopping center and the school, each of which are 2km apart. Is it possible
for them to built their houses such that each house is more than 1 km from the others?
[Hint: Divide the equilateral triangle into four regions!]

 

[pka]

Five Professors want to built their houses within the triangular area between the Uni-versity, the shopping center and the school, each of which are 2km apart. Is it possible for them to built their houses such that each house is more than 1 km from the others? [Hint: Divide the equilateral triangle into four regions!]

Draw the equilateral triangle. Join the three midpoints. Then there are four equilateral sub-triangles. Each has a side of 1km. Therefore no point two points on or interior to one of the four triangles can be more that 1km apart.

 

[Littlelad]

Thanks a million thats great