three towns are located at the vertices of an equilateral triangle. The towns are 8, 5, and 3 miles, respectively, from a store. How far apart are the towns?

I am at a quandary as to how to set up the solution for this one. I have it narrowed down to between 6.7 and 7.3 miles using Geo Sketchpad, but do not know how to solve algebraically for one value should it exist or to show the range works. Thanks,

 
I don't see a nice way to approach the problem, but a symbol manipulator such as Mathematica should be able to obtain a numerical solution. One can use the cosine law to obtain two equations in the unknown angles A and B, where A is the angle between the 5 and 8 mile paths, and B is the angle between the 8 and 3 mile paths. Let x be the side of the equilateral triangle.

x2
= 64 + 25 - 80 cos A
= 64 + 9 - 48 cos B
= 9 + 25 - 30 cos(A+B).

Subtract the second from the first and the third from the second to get,
16 = 80 cos A - 48 cos B
39 = 48 cos B - 30 cos (A+B)

Since you can change the second equation into an equation involving cos A and cos B (by expanding cos(A+B) and replacing the resulting sin A and sin B with square roots), Mathematica might be able to solve the two equations for cos A and cos B. Alternatively, One can use the first equation to get an expression for A in terms of cos B, and plug that in for A to get one horrific equation involving B.

Chris

I was a bit hasty in finding the side length of an equilateral triangle for which there is a point P whose distances from the vertices are 3, 5, and 8. Ed and I looked at it again and he noticed that P lies on the triangle's circcumcircle. That makes the problem easy! The side length is 7:

We are given an equilateral triangle TFE whose side has an unknown length x. There is a point P at distance 3 from T, 5 from F and 8 from E.

Step 1. Show that P lies on the circumcircle of TFE
We use Ptolemy's Theorem which says that if PT EF + FP TE = EP TF, then P, T, E, F lie on a circle in that order. So we check, 3x + 5x = 8x.

Step 2. Use the theorem that opposite angles of the cyclic quadrilateral PTEF add up to 180 degrees. Since the angle at E (namely FET) is 60 degrees, the angle at P (namely TPF) is 120 degrees.

Step 3. Apply the cosine law to triangle FPT (using cos 120 = -.5).
x2 = 32 + 52 - 2 3 5 (-.5) = 49. Thus x = 7.

Chris and Ed