Problem: Three towns in Ireland, Poole, Bray, and Alton, are located
8, 3, and 5 miles, repectively, from a store. How far apart are the towns
from one another if they are located at the vertices of an equilateral triangle?
Is the store inside, on, or outside the traingle?
We can then see that where the two circles intersect are three miles
and five miles away from Bray and Alton respectively. Now we can consider
if either or those points are 8mi away from Poole. We can look at the above
picture and elliminate point G because it is too close to Poole. With G
elliminated, we can then consider point E. Upon measurement we find that
the measure of line AE is 7.95cm.
This construction was done using computer software and the measurements
are accurate. Unfortunately, it is difficult to get lines to be the exact
length that you want them. Because of these minute differences in measurement
to lengths appear to be slightly off the mark.
This construction shows the point E to be 3 miles from Bray, 5 miles from
Alton, and 8 miles from Poole. With this restriction met, we can measure
the length between the cities and get 6.5 miles.
With the construction done here the store is outside the triangle. I did
try several constructions that might have the store inside or on the triangle,
but I was unsuccessful. That of course does not make it proof that there
are not solutions that exist for those situations, it simply means I am
not yet smart enough to find them!!