In George's small home town there are three churches, equidistant from each other as the crow flies - that...

geometry/three.churches New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

In George's small home town there are three churches, equidistant from
each other as the crow flies - that is to say, they stand at the
corners of an equilateral triangle.  George is standing exactly 1 mile
in a straight line from one church, 5/8 mile from the second church and
3/8 mile from the third.  Exactly how far apart are the three churches
from each other?

-- Peter Chapman

Show solution

Put the triangle on the coordinate plane, with the "5/8 mile church" at
(-a,0), the "3/8 mile church" at (a,0), and the "1 mile church" at
(0,a*sqrt(3)).  The distance between each pair of churches = 2a.
Let (x,y) be George's location:
(x+a)^2 + y^2 = (5/8)^2
(x-a)^2 + y^2 = (3/8)^2
x^2 + (y-sqrt(3)*a)^2 = 1^2
Subtracting the second equation from the first, x = 1/16a
Replacing x in the third equation:
1/(256a^2) + (y^2 - y*2a*sqrt(3) + 3a^2) = 1, or
y = (2a^2 - (47/64)) / (a*2*sqrt(3))
Replacing x and y in the first equation:
(a + (1/16a))^2 + ((a - 47/128a)^2)/3 = 25/64
which reduces to
(a^2 - (49/256))^2 = 0
Since a > 0 in this case, a = 7/16, and 2a = the distance between the
churches = 7/8.

-- Don Del Grande <del_gran@ix.netcom.com> (February 1998)