Henry Ernest Dudeney/Puzzles and Curious Problems/328 - Choosing a Site/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $328$

Choosing a Site
A man bought an estate enclosed by three straight roads forming an equilateral triangle.
He wished to build a house somewhere on the estate so that if he should have a straight drive from the front to each of the three roads,
he might be put to least expense.
Where should be build the house?


Solution

Anywhere he likes.


Proof

In Dudeney's words:

This was a little jest.


Let perpendiculars be drawn to the sides of an equilateral triangle from any point within it.

Then from Viviani's Theorem, the sum of their lengths is equal to the height of that equilateral triangle.

So, from whatever point you choose inside this equilateral triangle, the sum of those lengths is the same.

$\blacksquare$


Sources

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