Alice, Bob and Carol are standing on a straight line.

geometry/four.people New

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

Problem

Alice, Bob and Carol are standing on a straight line.  There's 1 meter
between Alice and Bob, Bob and David, David and Carol.  Alice is the
same distance from David as she is from Carol.  How far apart are Bob
and Carol?  Give all non zero answers.

-- Francois Grieu <fgrieu@micronet.fr>, wording by
   Chris_Lusby_Taylor@novell.com

Show solution

The possible distances can easily be computed as 0, (-1+sqrt(5))/2 and
(1+sqrt(5))/2 (if "A, B, C in a straight line" does not infer that B
has to be between A and C).

For if A(-1/0) and B(0/0), then C(c/0), D(d1/d2) with d1^2+d2^2=1,
(d1-c)^2+d2^2=1 and (d1+1)^2+d2^2=(c+1)^2. The first two equations give
c(c-2d1)=0, hence either c=0 or c=2d1. The first and third equation
yield 2d1+1=c^2+2c, which results in the golden ratio equation c^2+c-1=0
when 2d1 is replaced by c.

I'd like to know how this can be constructed using elementary geometry.

-- Eva Zerz <zerz@jupiter.mathematik.uni-kl.de>

1 Meter.  All persons were at the vertices of a tetrahedron with edges
of length 1 Meter.

-- John J. Vriezen <vriezen@millcomm.com>

Rather strange tetrahedron, if you ask me.

-- Seth Breidbart <sethb@panix.com>