(related to Problem: The Three Villages)
Calling the three villages by their initial letters, it is clear that the three roads form a triangle, $A,$ $B,$ $C,$ with a perpendicular, measuring twelve miles, dropped from $C$ to the base $A, B.$ This divides our triangle into two right-angled triangles with a twelve-mile side in common. It is then found that the distance from $A$ to $C$ is $15$ miles, from $C$ to $B$ $20$ miles, and from $A$ to $B$ $25$ (that is $9$ and $16$) miles. These figures are easily proved, for the square of $12$ added to the square of $9$ equals the square of $15,$ and the square of $12$ added to the square of $16$ equals the square of $20.$
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